Data.Number.Erf:$dmerfcx from erf-2.0.0.0

Percentage Accurate: 100.0% → 100.0%

Time: 34.7s

Alternatives: 17

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right)\\ \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(t\_0 \cdot t\_0\right), -1\right), \frac{1}{\mathsf{fma}\left(y \cdot y, t\_0, -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.16666666666666666 0.5)))
   (if (<= (* y y) 4e+141)
     (*
      x
      (fma
       (* (* y y) (fma (* y y) (* (* y y) (* t_0 t_0)) -1.0))
       (/ 1.0 (fma (* y y) t_0 -1.0))
       1.0))
     (* x (* y (* 0.5 (* y (* y y))))))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 0.5);
	double tmp;
	if ((y * y) <= 4e+141) {
		tmp = x * fma(((y * y) * fma((y * y), ((y * y) * (t_0 * t_0)), -1.0)), (1.0 / fma((y * y), t_0, -1.0)), 1.0);
	} else {
		tmp = x * (y * (0.5 * (y * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.16666666666666666, 0.5)
	tmp = 0.0
	if (Float64(y * y) <= 4e+141)
		tmp = Float64(x * fma(Float64(Float64(y * y) * fma(Float64(y * y), Float64(Float64(y * y) * Float64(t_0 * t_0)), -1.0)), Float64(1.0 / fma(Float64(y * y), t_0, -1.0)), 1.0));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 * Float64(y * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]}, If[LessEqual[N[(y * y), $MachinePrecision], 4e+141], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(y * y), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.00000000000000007e141

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right) + 1\right) \cdot \left(y \cdot y\right)} + 1\right) \]

      2. flip-+ N/A

         \[\leadsto x \cdot \left(\color{blue}{\frac{\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) - 1 \cdot 1}{y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right) - 1}} \cdot \left(y \cdot y\right) + 1\right) \]

      3. associate-*l/ N/A

         \[\leadsto x \cdot \left(\color{blue}{\frac{\left(\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) - 1 \cdot 1\right) \cdot \left(y \cdot y\right)}{y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right) - 1}} + 1\right) \]

      4. div-inv N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) - 1 \cdot 1\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right) - 1}} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(\left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right)\right) - 1 \cdot 1\right) \cdot \left(y \cdot y\right), \frac{1}{y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right) - 1}, 1\right)} \]

   7. Applied egg-rr 92.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right)\right), -1\right) \cdot \left(y \cdot y\right), \frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), -1\right)}, 1\right)} \]

   if 4.00000000000000007e141 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot {y}^{4}\right) \cdot \frac{1}{2}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left({y}^{4} \cdot \frac{1}{2}\right)} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

          \[\leadsto x \cdot \left(y \cdot \left(\frac{1}{2} \cdot \color{blue}{{y}^{3}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{3}\right)}\right) \]

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right)\right), -1\right), \frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), -1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

   9. +-inverses N/A

      \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   16. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   17. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   20. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   21. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

   22. flip-+ N/A

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

   23. count-2 N/A

       \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

   24. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{2 \cdot y}{2}}} \]

4. Applied egg-rr 75.8%

   \[\leadsto x \cdot e^{\color{blue}{y}} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{-6 + \frac{18}{y \cdot y}}\right), 1\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (fma
   (* y y)
   (fma y (* y (* (* y y) (/ -1.0 (+ -6.0 (/ 18.0 (* y y)))))) 1.0)
   1.0)))

C

double code(double x, double y) {
	return x * fma((y * y), fma(y, (y * ((y * y) * (-1.0 / (-6.0 + (18.0 / (y * y)))))), 1.0), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(Float64(y * y), fma(y, Float64(y * Float64(Float64(y * y) * Float64(-1.0 / Float64(-6.0 + Float64(18.0 / Float64(y * y)))))), 1.0), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(-1.0 / N[(-6.0 + N[(18.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]

5. Simplified 93.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\frac{\left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}{y \cdot \left(y \cdot \frac{1}{6}\right) - \frac{1}{2}}}, 1\right), 1\right) \]

   2. clear-num N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\frac{1}{\frac{y \cdot \left(y \cdot \frac{1}{6}\right) - \frac{1}{2}}{\left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}}}, 1\right), 1\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\frac{1}{\frac{y \cdot \left(y \cdot \frac{1}{6}\right) - \frac{1}{2}}{\left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}}}, 1\right), 1\right) \]

   4. clear-num N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\color{blue}{\frac{1}{\frac{\left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}{y \cdot \left(y \cdot \frac{1}{6}\right) - \frac{1}{2}}}}}, 1\right), 1\right) \]

   5. flip-+ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{1}{\color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}}}}, 1\right), 1\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\color{blue}{\frac{1}{y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}}}}, 1\right), 1\right) \]

   7. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{1}{\color{blue}{\left(y \cdot y\right) \cdot \frac{1}{6}} + \frac{1}{2}}}, 1\right), 1\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{6}, \frac{1}{2}\right)}}}, 1\right), 1\right) \]

   9. *-lowering-*.f64 93.4

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{1}{\mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 0.5\right)}}, 1\right), 1\right) \]

7. Applied egg-rr 93.4%

   \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right)}}}, 1\right), 1\right) \]

8. Taylor expanded in y around inf

   \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\color{blue}{\frac{6 - 18 \cdot \frac{1}{{y}^{2}}}{{y}^{2}}}}, 1\right), 1\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\color{blue}{\frac{6 - 18 \cdot \frac{1}{{y}^{2}}}{{y}^{2}}}}, 1\right), 1\right) \]

   2. sub-neg N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{\color{blue}{6 + \left(\mathsf{neg}\left(18 \cdot \frac{1}{{y}^{2}}\right)\right)}}{{y}^{2}}}, 1\right), 1\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{\color{blue}{6 + \left(\mathsf{neg}\left(18 \cdot \frac{1}{{y}^{2}}\right)\right)}}{{y}^{2}}}, 1\right), 1\right) \]

   4. associate-*r/ N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \left(\mathsf{neg}\left(\color{blue}{\frac{18 \cdot 1}{{y}^{2}}}\right)\right)}{{y}^{2}}}, 1\right), 1\right) \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \left(\mathsf{neg}\left(\frac{\color{blue}{18}}{{y}^{2}}\right)\right)}{{y}^{2}}}, 1\right), 1\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \color{blue}{\frac{\mathsf{neg}\left(18\right)}{{y}^{2}}}}{{y}^{2}}}, 1\right), 1\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \color{blue}{\frac{\mathsf{neg}\left(18\right)}{{y}^{2}}}}{{y}^{2}}}, 1\right), 1\right) \]

   8. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \frac{\color{blue}{-18}}{{y}^{2}}}{{y}^{2}}}, 1\right), 1\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \frac{-18}{\color{blue}{y \cdot y}}}{{y}^{2}}}, 1\right), 1\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \frac{-18}{\color{blue}{y \cdot y}}}{{y}^{2}}}, 1\right), 1\right) \]

   11. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \frac{-18}{y \cdot y}}{\color{blue}{y \cdot y}}}, 1\right), 1\right) \]

   12. *-lowering-*.f64 93.4

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\frac{6 + \frac{-18}{y \cdot y}}{\color{blue}{y \cdot y}}}, 1\right), 1\right) \]

10. Simplified 93.4%

    \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{\color{blue}{\frac{6 + \frac{-18}{y \cdot y}}{y \cdot y}}}, 1\right), 1\right) \]
11. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{6 + \frac{-18}{y \cdot y}} \cdot \left(y \cdot y\right)\right)}, 1\right), 1\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{6 + \frac{-18}{y \cdot y}} \cdot \left(y \cdot y\right)\right)}, 1\right), 1\right) \]

    3. frac-2neg N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(6 + \frac{-18}{y \cdot y}\right)\right)}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    4. metadata-eval N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(6 + \frac{-18}{y \cdot y}\right)\right)} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\left(6 + \frac{-18}{y \cdot y}\right)\right)}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    6. distribute-neg-in N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{\color{blue}{\left(\mathsf{neg}\left(6\right)\right) + \left(\mathsf{neg}\left(\frac{-18}{y \cdot y}\right)\right)}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    7. frac-2neg N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{\left(\mathsf{neg}\left(6\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(-18\right)}{\mathsf{neg}\left(y \cdot y\right)}}\right)\right)} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    8. distribute-frac-neg2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{\left(\mathsf{neg}\left(6\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(-18\right)}{y \cdot y}\right)\right)}\right)\right)} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    9. remove-double-neg N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{\left(\mathsf{neg}\left(6\right)\right) + \color{blue}{\frac{\mathsf{neg}\left(-18\right)}{y \cdot y}}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{\color{blue}{\left(\mathsf{neg}\left(6\right)\right) + \frac{\mathsf{neg}\left(-18\right)}{y \cdot y}}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    11. metadata-eval N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{\color{blue}{-6} + \frac{\mathsf{neg}\left(-18\right)}{y \cdot y}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{-6 + \color{blue}{\frac{\mathsf{neg}\left(-18\right)}{y \cdot y}}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    13. metadata-eval N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{-6 + \frac{\color{blue}{18}}{y \cdot y}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    14. *-lowering-*.f64 N/A

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{-6 + \frac{18}{\color{blue}{y \cdot y}}} \cdot \left(y \cdot y\right)\right), 1\right), 1\right) \]

    15. *-lowering-*.f64 93.4

        \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{-1}{-6 + \frac{18}{y \cdot y}} \cdot \color{blue}{\left(y \cdot y\right)}\right), 1\right), 1\right) \]

12. Applied egg-rr 93.4%

    \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{-1}{-6 + \frac{18}{y \cdot y}} \cdot \left(y \cdot y\right)\right)}, 1\right), 1\right) \]

13. Final simplification 93.4%

    \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{-6 + \frac{18}{y \cdot y}}\right), 1\right), 1\right) \]
14. Add Preprocessing

Alternative 5: 94.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  x
  (fma (* y y) (fma y (* y (fma y (* y 0.16666666666666666) 0.5)) 1.0) 1.0)))

C

double code(double x, double y) {
	return x * fma((y * y), fma(y, (y * fma(y, (y * 0.16666666666666666), 0.5)), 1.0), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(y, Float64(y * 0.16666666666666666), 0.5)), 1.0), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]

5. Simplified 93.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
6. Add Preprocessing

Alternative 6: 90.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.5, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-8)
   (* x (fma y y 1.0))
   (* x (* y (* y (fma y (* y 0.5) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = x * (y * (y * fma(y, (y * 0.5), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(x * Float64(y * Float64(y * fma(y, Float64(y * 0.5), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(y * N[(y * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 84.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{2} + \frac{1}{{y}^{2}}\right)\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 84.1

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

   8. Simplified 84.1%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.5, 1\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right), 1\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (fma (* y y) (fma y (* y (* (* y y) 0.16666666666666666)) 1.0) 1.0)))

C

double code(double x, double y) {
	return x * fma((y * y), fma(y, (y * ((y * y) * 0.16666666666666666)), 1.0), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(Float64(y * y), fma(y, Float64(y * Float64(Float64(y * y) * 0.16666666666666666)), 1.0), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]

5. Simplified 93.4%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}, 1\right), 1\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}, 1\right), 1\right) \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 93.4

      \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right), 1\right), 1\right) \]

8. Simplified 93.4%

   \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}, 1\right), 1\right) \]

9. Final simplification 93.4%

   \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right), 1\right), 1\right) \]
10. Add Preprocessing

Alternative 8: 90.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-8) (* x (fma y y 1.0)) (* x (* y (* 0.5 (* y (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = x * (y * (0.5 * (y * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 * Float64(y * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 84.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot {y}^{4}\right) \cdot \frac{1}{2}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left({y}^{4} \cdot \frac{1}{2}\right)} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. unpow3 N/A

          \[\leadsto x \cdot \left(y \cdot \left(\frac{1}{2} \cdot \color{blue}{{y}^{3}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{3}\right)}\right) \]

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. unpow2 N/A

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

      21. *-lowering-*.f64 84.0

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

   8. Simplified 84.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. distribute-rgt-in N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-in N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

5. Simplified 92.2%

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

Alternative 10: 90.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. distribute-rgt-in N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-in N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

5. Simplified 92.2%

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

6. Taylor expanded in y around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 91.9

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

8. Simplified 91.9%

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

Alternative 11: 68.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), x\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma x (* y (fma y (fma y 0.16666666666666666 0.5) 1.0)) x))

C

double code(double x, double y) {
	return fma(x, (y * fma(y, fma(y, 0.16666666666666666, 0.5), 1.0)), x);
}

Julia

function code(x, y)
	return fma(x, Float64(y * fma(y, fma(y, 0.16666666666666666, 0.5), 1.0)), x)
end

Wolfram

code[x_, y_] := N[(x * N[(y * N[(y * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

   9. +-inverses N/A

      \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   16. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   17. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   20. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   21. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

   22. flip-+ N/A

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

   23. count-2 N/A

       \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

   24. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{2 \cdot y}{2}}} \]

4. Applied egg-rr 75.8%

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

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left(x \cdot y\right) + \frac{1}{2} \cdot x\right)\right)} \]

7. Simplified 71.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), x\right)} \]
8. Add Preprocessing

Alternative 12: 80.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 5e-8) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d-8) then
        tmp = x
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-8:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = x;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-8)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 99.3%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-lft-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 69.5

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

   5. Simplified 69.5%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 69.5

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

   8. Simplified 69.5%

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

Alternative 13: 56.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 5e-8) x (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d-8) then
        tmp = x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-8:
		tmp = x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = x;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-8)
		tmp = x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], x, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 99.3%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-out N/A

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

      6. div-inv N/A

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

      7. div-inv N/A

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

      8. flip-+ N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

      9. +-inverses N/A

         \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

      10. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

      11. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

      14. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

      15. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

      16. distribute-lft-out N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

      17. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

      18. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

      19. +-inverses N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

      20. difference-of-squares N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

      21. +-inverses N/A

          \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

      22. flip-+ N/A

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

      23. count-2 N/A

          \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

      24. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{2 \cdot y}{2}}} \]

   4. Applied egg-rr 52.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 16.5

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

   7. Simplified 16.5%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 16.5

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

   10. Simplified 16.5%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.1% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-rgt-identity N/A

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

   2. distribute-lft-in N/A

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

   3. *-lowering-*.f64 N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. accelerator-lowering-fma.f64 85.1

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

5. Simplified 85.1%

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

Alternative 15: 56.0% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

   9. +-inverses N/A

      \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   16. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   17. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   20. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   21. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

   22. flip-+ N/A

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

   23. count-2 N/A

       \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

   24. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{2 \cdot y}{2}}} \]

4. Applied egg-rr 75.8%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 58.4

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

7. Simplified 58.4%

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

Alternative 16: 56.0% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

   9. +-inverses N/A

      \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   16. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   17. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   20. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   21. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

   22. flip-+ N/A

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

   23. count-2 N/A

       \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

   24. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{2 \cdot y}{2}}} \]

4. Applied egg-rr 75.8%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 58.4

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

7. Simplified 58.4%

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

Alternative 17: 51.1% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 52.8%

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

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (pow (exp y) y)))

  (* x (exp (* y y))))