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

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 11

Speedup: 1.0×

• 49.7% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Power[N[Exp[y], $MachinePrecision], y], $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. *-commutative N/A

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

   3. lower-*.f64 100.0

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

   4. lift-exp.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 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. lower-exp.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 100.0

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

6. Applied rewrites 100.0%

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

Alternative 3: 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 4: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Power[N[(1.0 + y), $MachinePrecision], y], $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. *-commutative N/A

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

   3. lower-*.f64 100.0

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

   4. lift-exp.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

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

   1. lower-+.f64 76.4

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

7. Applied rewrites 76.4%

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

Alternative 5: 94.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. distribute-lft-in N/A

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

   7. +-commutative N/A

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

5. Applied rewrites 90.6%

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

7. Applied rewrites 92.8%

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

Alternative 6: 94.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. distribute-lft-in N/A

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

   7. +-commutative N/A

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

5. Applied rewrites 90.6%

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

7. Applied rewrites 92.8%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 92.7%

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

Alternative 7: 91.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. 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

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

7. Applied rewrites 86.8%

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

9. Applied rewrites 90.5%

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

10. Final simplification 90.5%

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

Alternative 8: 91.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. distribute-lft-in N/A

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

   7. +-commutative N/A

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

5. Applied rewrites 90.6%

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

7. Applied rewrites 92.8%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 90.5%

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

Alternative 9: 82.3% 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 100.0

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

   4. lift-exp.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
6. 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 82.5

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

7. Applied rewrites 82.5%

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

8. Final simplification 82.5%

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

Alternative 10: 75.8% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lower-*.f64 100.0

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

   4. lift-exp.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
6. 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 82.5

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

7. Applied rewrites 82.5%

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

9. Applied rewrites 78.5%

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

10. Final simplification 78.5%

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

Alternative 11: 51.5% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * 1.0), $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 49.3%

   \[\leadsto x \cdot \color{blue}{1} \]
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 2024320 
(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))))