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

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 10

Speedup: 1.0×

• 51.7% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 93.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
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 93.0%

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

7. Applied rewrites 93.7%

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

Alternative 3: 93.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
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 93.0%

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

7. Applied rewrites 93.7%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 93.5%

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

Alternative 4: 92.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
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 93.0%

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

7. Applied rewrites 93.0%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 92.7%

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

Alternative 5: 90.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
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 93.0%

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

7. Applied rewrites 93.7%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 92.5%

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

Alternative 6: 90.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
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 93.0%

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

7. Applied rewrites 93.7%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 92.5%

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

12. Applied rewrites 92.5%

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

Alternative 7: 75.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d-5) then
        tmp = x * 1.0d0
    else
        tmp = (x * y) * y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-5:
		tmp = x * 1.0
	else:
		tmp = (x * y) * y
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-5)
		tmp = x * 1.0;
	else
		tmp = (x * y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.00000000000000024e-5

   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 98.3%

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

   if 5.00000000000000024e-5 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
   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 85.7%

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 45.1

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

   10. Applied rewrites 45.1%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 45.1%

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

Alternative 8: 81.3% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 80.8

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

5. Applied rewrites 80.8%

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

Alternative 9: 75.4% 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. 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 93.0%

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

7. Applied rewrites 93.7%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 72.7

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

10. Applied rewrites 72.7%

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

Alternative 10: 49.5% 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 51.7%

   \[\leadsto x \cdot \color{blue}{1} \]
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 2024326 
(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))))