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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 5

Speedup: 1.0×

• 49.6% 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

\[\begin{array}{l} \\ x \cdot {e}^{\left({y}^{2}\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (pow E (pow y 2.0))))

C

double code(double x, double y) {
	return x * pow(((double) M_E), pow(y, 2.0));
}

Java

public static double code(double x, double y) {
	return x * Math.pow(Math.E, Math.pow(y, 2.0));
}

Python

def code(x, y):
	return x * math.pow(math.e, math.pow(y, 2.0))

Julia

function code(x, y)
	return Float64(x * (exp(1) ^ (y ^ 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (2.71828182845904523536 ^ (y ^ 2.0));
end

Wolfram

code[x_, y_] := N[(x * N[Power[E, N[Power[y, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

   3. exp-1-e 100.0%

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

   4. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 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. Final simplification 100.0%

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

Alternative 3: 57.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) + -1}{1 + \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.0) x (/ (+ (* (+ x 1.0) (+ x 1.0)) -1.0) (+ 1.0 (+ x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.0) {
		tmp = x;
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) + -1.0) / (1.0 + (x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.0d0) then
        tmp = x
    else
        tmp = (((x + 1.0d0) * (x + 1.0d0)) + (-1.0d0)) / (1.0d0 + (x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.0) {
		tmp = x;
	} else {
		tmp = (((x + 1.0) * (x + 1.0)) + -1.0) / (1.0 + (x + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.0:
		tmp = x
	else:
		tmp = (((x + 1.0) * (x + 1.0)) + -1.0) / (1.0 + (x + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.0)
		tmp = x;
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) + -1.0) / Float64(1.0 + Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.0)
		tmp = x;
	else
		tmp = (((x + 1.0) * (x + 1.0)) + -1.0) / (1.0 + (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.0], x, N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.9%

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

   if 8 < y

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot y\right)\right)}} \]

      2. expm1-udef 100.0%

         \[\leadsto x \cdot e^{\color{blue}{e^{\mathsf{log1p}\left(y \cdot y\right)} - 1}} \]

      3. exp-diff 100.0%

         \[\leadsto x \cdot \color{blue}{\frac{e^{e^{\mathsf{log1p}\left(y \cdot y\right)}}}{e^{1}}} \]

      4. log1p-udef 100.0%

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

      5. rem-exp-log 100.0%

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

      6. pow2 100.0%

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

      7. exp-1-e 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto x \cdot \color{blue}{\frac{e^{1 + {y}^{2}}}{e}} \]
   5. Step-by-step derivation

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

      3. add-exp-log 100.0%

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

      4. clear-num 100.0%

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

      5. e-exp-1 100.0%

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

      6. div-exp 100.0%

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

      7. add-exp-log 100.0%

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

      8. log1p-udef 100.0%

         \[\leadsto \frac{x}{e^{\log \left(\frac{1}{e^{e^{\color{blue}{\mathsf{log1p}\left({y}^{2}\right)}} - 1}}\right)}} \]

      9. expm1-udef 100.0%

         \[\leadsto \frac{x}{e^{\log \left(\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2}\right)\right)}}}\right)}} \]

      10. expm1-log1p-u 100.0%

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

      11. *-un-lft-identity 100.0%

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

      12. pow-exp 100.0%

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

      13. e-exp-1 100.0%

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

      14. neg-log 100.0%

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

      15. e-exp-1 100.0%

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

      16. pow-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{-{y}^{2}}}{x}}} \]

      2. inv-pow 100.0%

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

      3. add-exp-log 38.7%

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

      4. div-exp 38.7%

         \[\leadsto {\color{blue}{\left(e^{\left(-{y}^{2}\right) - \log x}\right)}}^{-1} \]

   8. Applied egg-rr 4.4%

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

      1. unpow-1 4.4%

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

      2. remove-double-div 4.4%

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

      3. add-exp-log 2.0%

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

      4. add-sqr-sqrt 1.7%

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

      5. sqrt-unprod 2.2%

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

      6. sqr-neg 2.2%

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

      7. log-rec 2.2%

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

      8. log-rec 2.2%

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

      9. sqrt-unprod 0.5%

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

      10. add-sqr-sqrt 1.1%

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

      11. expm1-log1p-u 1.1%

          \[\leadsto e^{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x}\right)\right)\right)}} \]

      12. expm1-udef 1.4%

          \[\leadsto e^{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1\right)}} \]

      13. add-exp-log 1.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1} \]

      14. flip-- 7.5%

          \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot e^{\mathsf{log1p}\left(\frac{1}{x}\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{1}{x}\right)} + 1}} \]

   10. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\frac{\left(1 + x\right) \cdot \left(1 + x\right) - 1}{\left(1 + x\right) + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot \left(x + 1\right) + -1}{1 + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 2.7% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto x \cdot e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot y\right)\right)}} \]

   2. expm1-udef 100.0%

      \[\leadsto x \cdot e^{\color{blue}{e^{\mathsf{log1p}\left(y \cdot y\right)} - 1}} \]

   3. exp-diff 100.0%

      \[\leadsto x \cdot \color{blue}{\frac{e^{e^{\mathsf{log1p}\left(y \cdot y\right)}}}{e^{1}}} \]

   4. log1p-udef 100.0%

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

   5. rem-exp-log 100.0%

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

   6. pow2 100.0%

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

   7. exp-1-e 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{\frac{e^{1 + {y}^{2}}}{e}} \]
5. Step-by-step derivation

   1. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

   3. add-exp-log 100.0%

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

   4. clear-num 100.0%

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

   5. e-exp-1 100.0%

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

   6. div-exp 100.0%

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

   7. add-exp-log 100.0%

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

   8. log1p-udef 100.0%

      \[\leadsto \frac{x}{e^{\log \left(\frac{1}{e^{e^{\color{blue}{\mathsf{log1p}\left({y}^{2}\right)}} - 1}}\right)}} \]

   9. expm1-udef 100.0%

      \[\leadsto \frac{x}{e^{\log \left(\frac{1}{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2}\right)\right)}}}\right)}} \]

   10. expm1-log1p-u 100.0%

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

   11. *-un-lft-identity 100.0%

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

   12. pow-exp 100.0%

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

   13. e-exp-1 100.0%

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

   14. neg-log 100.0%

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

   15. e-exp-1 100.0%

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

   16. pow-exp 100.0%

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

6. Applied egg-rr 100.0%

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

   1. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{-{y}^{2}}}{x}}} \]

   2. inv-pow 99.8%

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

   3. add-exp-log 48.6%

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

   4. div-exp 48.6%

      \[\leadsto {\color{blue}{\left(e^{\left(-{y}^{2}\right) - \log x}\right)}}^{-1} \]

8. Applied egg-rr 51.6%

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

   1. unpow-1 51.6%

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

   2. add-sqr-sqrt 28.5%

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

   3. associate-/r* 28.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x}}}} \]

   4. metadata-eval 28.5%

      \[\leadsto \frac{1}{\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}}{\sqrt{x}}} \]

   5. sqrt-div 28.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{x}}}}{\sqrt{x}}} \]

   6. clear-num 28.6%

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{\sqrt{\frac{1}{x}}}} \]

   7. pow1/2 28.6%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{{\left(\frac{1}{x}\right)}^{0.5}}} \]

   8. pow1 28.6%

      \[\leadsto \frac{\sqrt{x}}{{\color{blue}{\left({\left(\frac{1}{x}\right)}^{1}\right)}}^{0.5}} \]

   9. pow-to-exp 26.8%

      \[\leadsto \frac{\sqrt{x}}{{\color{blue}{\left(e^{\log \left(\frac{1}{x}\right) \cdot 1}\right)}}^{0.5}} \]

   10. metadata-eval 26.8%

       \[\leadsto \frac{\sqrt{x}}{{\left(e^{\log \left(\frac{1}{x}\right) \cdot 1}\right)}^{\color{blue}{\left(-1 \cdot -0.5\right)}}} \]

   11. metadata-eval 26.8%

       \[\leadsto \frac{\sqrt{x}}{{\left(e^{\log \left(\frac{1}{x}\right) \cdot 1}\right)}^{\left(-1 \cdot \color{blue}{\frac{-1}{2}}\right)}} \]

   12. pow-exp 26.8%

       \[\leadsto \frac{\sqrt{x}}{\color{blue}{e^{\left(\log \left(\frac{1}{x}\right) \cdot 1\right) \cdot \left(-1 \cdot \frac{-1}{2}\right)}}} \]

   13. *-rgt-identity 26.8%

       \[\leadsto \frac{\sqrt{x}}{e^{\color{blue}{\log \left(\frac{1}{x}\right)} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   14. add-sqr-sqrt 12.7%

       \[\leadsto \frac{\sqrt{x}}{e^{\color{blue}{\left(\sqrt{\log \left(\frac{1}{x}\right)} \cdot \sqrt{\log \left(\frac{1}{x}\right)}\right)} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   15. sqrt-unprod 14.1%

       \[\leadsto \frac{\sqrt{x}}{e^{\color{blue}{\sqrt{\log \left(\frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   16. log-rec 14.1%

       \[\leadsto \frac{\sqrt{x}}{e^{\sqrt{\color{blue}{\left(-\log x\right)} \cdot \log \left(\frac{1}{x}\right)} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   17. log-rec 14.1%

       \[\leadsto \frac{\sqrt{x}}{e^{\sqrt{\left(-\log x\right) \cdot \color{blue}{\left(-\log x\right)}} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   18. sqr-neg 14.1%

       \[\leadsto \frac{\sqrt{x}}{e^{\sqrt{\color{blue}{\log x \cdot \log x}} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   19. sqrt-unprod 1.2%

       \[\leadsto \frac{\sqrt{x}}{e^{\color{blue}{\left(\sqrt{\log x} \cdot \sqrt{\log x}\right)} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   20. add-sqr-sqrt 2.3%

       \[\leadsto \frac{\sqrt{x}}{e^{\color{blue}{\log x} \cdot \left(-1 \cdot \frac{-1}{2}\right)}} \]

   21. pow-to-exp 2.3%

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

   22. metadata-eval 2.3%

       \[\leadsto \frac{\sqrt{x}}{{x}^{\left(-1 \cdot \color{blue}{-0.5}\right)}} \]

10. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{\frac{\sqrt{x}}{\sqrt{x}}} \]
11. Step-by-step derivation

    1. *-inverses 2.8%

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

12. Simplified 2.8%

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

13. Final simplification 2.8%

    \[\leadsto 1 \]
14. Add Preprocessing

Alternative 5: 51.5% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 51.7%

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

4. Final simplification 51.7%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 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 2024036 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :herbie-target
  (* x (pow (exp y) y))

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