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

Average Accuracy: 99.9% → 99.9%

Time: 4.4s

Precision: binary64

Cost: 19584

Math

\[x \cdot e^{y \cdot y} \]

↓

\[x \cdot {\left(\sqrt{e^{y}}\right)}^{\left(y + y\right)} \]

FPCore

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

↓

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

C

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

↓

double code(double x, double y) {
	return x * pow(sqrt(exp(y)), (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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	99.9%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 99.9%

   \[x \cdot e^{y \cdot y} \]

2. Simplified 99.9%

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

   [Start] 99.9	\[ x \cdot e^{y \cdot y} \]
   exp-prod [=>] 99.9	\[ x \cdot \color{blue}{{\left(e^{y}\right)}^{y}} \]

3. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ x \cdot {\left(e^{y}\right)}^{y} \]
   add-sqr-sqrt [=>] 99.9	\[ x \cdot {\color{blue}{\left(\sqrt{e^{y}} \cdot \sqrt{e^{y}}\right)}}^{y} \]
   unpow-prod-down [=>] 99.9	\[ x \cdot \color{blue}{\left({\left(\sqrt{e^{y}}\right)}^{y} \cdot {\left(\sqrt{e^{y}}\right)}^{y}\right)} \]

4. Simplified 99.9%

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

   [Start] 99.9	\[ x \cdot \left({\left(\sqrt{e^{y}}\right)}^{y} \cdot {\left(\sqrt{e^{y}}\right)}^{y}\right) \]
   pow-sqr [=>] 99.9	\[ x \cdot \color{blue}{{\left(\sqrt{e^{y}}\right)}^{\left(2 \cdot y\right)}} \]
   count-2 [<=] 99.9	\[ x \cdot {\left(\sqrt{e^{y}}\right)}^{\color{blue}{\left(y + y\right)}} \]

5. Final simplification 99.9%

   \[\leadsto x \cdot {\left(\sqrt{e^{y}}\right)}^{\left(y + y\right)} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13056

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

Alternative 2
Accuracy	99.9%
Cost	6720

\[x \cdot e^{y \cdot y} \]

Alternative 3
Accuracy	99.1%
Cost	448

\[x \cdot \left(y \cdot y + 1\right) \]

Alternative 4
Accuracy	99.1%
Cost	448

\[x + x \cdot \left(y \cdot y\right) \]

Alternative 5
Accuracy	98.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))))