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

Percentage Accurate: 100.0% → 100.0%

Time: 43.2s

Alternatives: 11

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.4890192925044873e-177
  2. y = 1.8353711055128213e+62

• 24.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400000:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -400000.0)
   (exp (* x y))
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -400000.0) {
		tmp = exp((x * y));
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -400000.0)
		tmp = exp(Float64(x * y));
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -400000.0], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e5

   1. Initial program 100.0%

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

   4. Applied rewrites 43.6%

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

   if -4e5 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400000:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -400000.0)
   (exp x)
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -400000.0) {
		tmp = exp(x);
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -400000.0)
		tmp = exp(x);
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -400000.0], N[Exp[x], $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e5

   1. Initial program 100.0%

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

   4. Applied rewrites 67.3%

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

   if -4e5 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

5. Applied rewrites 65.0%

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

Alternative 5: 70.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.1)
   (fma x (* y y) 1.0)
   (* x (* 0.5 (* x (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.1) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = x * (0.5 * (x * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 0.1)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 0.1], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.10000000000000001

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if 0.10000000000000001 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. rem-exp-log N/A

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

      5. log-prod N/A

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

      6. exp-sum N/A

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

      7. rem-exp-log N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-log.f64 80.9

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 83.9

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

   10. Applied rewrites 83.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot \left(x \cdot 0.5\right), x\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-92)
   (fma x (* y y) 1.0)
   (if (<= y 1.56e+98)
     (fma y (fma y (* x (* x 0.5)) x) 1.0)
     (fma y (fma y (fma y 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-92) {
		tmp = fma(x, (y * y), 1.0);
	} else if (y <= 1.56e+98) {
		tmp = fma(y, fma(y, (x * (x * 0.5)), x), 1.0);
	} else {
		tmp = fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-92)
		tmp = fma(x, Float64(y * y), 1.0);
	elseif (y <= 1.56e+98)
		tmp = fma(y, fma(y, Float64(x * Float64(x * 0.5)), x), 1.0);
	else
		tmp = fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e-92], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 1.56e+98], N[(y * N[(y * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision], N[(y * N[(y * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.30000000000000016e-92

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if 2.30000000000000016e-92 < y < 1.56e98

   1. Initial program 100.0%

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

   4. Applied rewrites 92.1%

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lift-exp.f64 N/A

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

      4. lower-pow.f64 67.6

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

   6. Applied rewrites 67.6%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.6

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

   9. Applied rewrites 67.6%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 52.8

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

   12. Applied rewrites 52.8%

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

   if 1.56e98 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 46.8

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

   7. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.0% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

5. Applied rewrites 63.7%

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

Alternative 8: 66.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.56 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.56e+98)
   (fma x (* y y) 1.0)
   (fma y (fma y (fma y 0.16666666666666666 0.5) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.56e+98) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.56e+98)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.56e+98], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(y * N[(y * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.56e98

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 1.56e98 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 46.8

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

   7. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.1) 1.0 (fma x y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.1) {
		tmp = 1.0;
	} else {
		tmp = fma(x, y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 0.1)
		tmp = 1.0;
	else
		tmp = fma(x, y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 0.1], 1.0, N[(x * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.10000000000000001

   1. Initial program 100.0%

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

   4. Applied rewrites 57.5%

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

   if 0.10000000000000001 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 14.7

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

   7. Applied rewrites 14.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.8% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * 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. 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. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 59.1

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

5. Applied rewrites 59.1%

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

Alternative 11: 50.6% 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

4. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))