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

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 11

Speedup: 1.0×

• 50.2% 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, 0.9× speedup

Math

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

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(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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

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

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

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

   6. sinh-cosh N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-exp.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 3: 94.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * y), $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. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

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

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

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

   6. sinh-cosh N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-exp.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 92.2%

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

9. Applied rewrites 92.9%

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

11. Applied rewrites 94.4%

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

Alternative 4: 90.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * N[(N[(0.5 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * y), $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. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

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

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

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

   6. sinh-cosh N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-exp.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 89.2%

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

9. Applied rewrites 92.2%

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

Alternative 5: 89.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

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

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

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

   6. sinh-cosh N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-exp.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 89.2%

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

9. Applied rewrites 89.9%

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

10. Taylor expanded in y around inf

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

12. Applied rewrites 89.6%

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

Alternative 6: 87.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

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

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

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

   6. sinh-cosh N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-exp.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 89.2%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 88.9%

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

Alternative 7: 82.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e+27) (fma (* y x) y x) (* (* y y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+27) {
		tmp = fma((y * x), y, x);
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 4e+27)
		tmp = fma(Float64(y * x), y, x);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4e+27], N[(N[(y * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.0000000000000001e27

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 95.0%

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

   if 4.0000000000000001e27 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 68.4%

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

Alternative 8: 81.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e-7) (* 1.0 x) (* (* y y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-7) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * y) * x;
	}
	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) <= 2d-7) then
        tmp = 1.0d0 * x
    else
        tmp = (y * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-7) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-7:
		tmp = 1.0 * x
	else:
		tmp = (y * y) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-7)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e-7)
		tmp = 1.0 * x;
	else
		tmp = (y * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-7], N[(1.0 * x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.4%

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

   if 1.9999999999999999e-7 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * 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 <= 1.0d0) then
        tmp = 1.0d0 * x
    else
        tmp = (y * x) * y
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 69.3%

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

   if 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.7%

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

   10. Applied rewrites 44.5%

       \[\leadsto \left(y \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

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

Alternative 10: 82.0% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * x + x), $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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 82.8

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

5. Applied rewrites 82.8%

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

Alternative 11: 51.0% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 52.6%

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

6. Final simplification 52.6%

   \[\leadsto 1 \cdot x \]
7. 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 2024298 
(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))))