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

Average Error: 0.0 → 0.0

Time: 6.8s

Precision: binary64

Cost: 13440

Math

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

↓

\[x \cdot {\left(e^{y \cdot 4}\right)}^{\left(2 \cdot \frac{y}{8}\right)} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (* x (pow (exp (* y 4.0)) (* 2.0 (/ y 8.0)))))

C

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

↓

double code(double x, double y) {
	return x * pow(exp((y * 4.0)), (2.0 * (y / 8.0)));
}

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 * (exp((y * 4.0d0)) ** (2.0d0 * (y / 8.0d0)))
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.exp((y * 4.0)), (2.0 * (y / 8.0)));
}

Python

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

↓

def code(x, y):
	return x * math.pow(math.exp((y * 4.0)), (2.0 * (y / 8.0)))

Julia

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

↓

function code(x, y)
	return Float64(x * (exp(Float64(y * 4.0)) ^ Float64(2.0 * Float64(y / 8.0))))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = x * (exp((y * 4.0)) ^ (2.0 * (y / 8.0)));
end

Wolfram

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

↓

code[x_, y_] := N[(x * N[Power[N[Exp[N[(y * 4.0), $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(y / 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ x \cdot e^{y \cdot y} \]
   exp-prod [=>] 0.0	\[ x \cdot \color{blue}{{\left(e^{y}\right)}^{y}} \]

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

   \[\leadsto x \cdot \color{blue}{{\left({\left(e^{y}\right)}^{2}\right)}^{\left(y \cdot 0.5\right)}} \]

5. Applied egg-rr 0.0

   \[\leadsto x \cdot \color{blue}{\left({\left(e^{y \cdot 4}\right)}^{\left(\frac{y}{8}\right)} \cdot {\left(e^{y \cdot 4}\right)}^{\left(\frac{y}{8}\right)}\right)} \]

6. Simplified 0.0

   \[\leadsto x \cdot \color{blue}{{\left(e^{y \cdot 4}\right)}^{\left(2 \cdot \frac{y}{8}\right)}} \]
   Proof

   [Start] 0.0	\[ x \cdot \left({\left(e^{y \cdot 4}\right)}^{\left(\frac{y}{8}\right)} \cdot {\left(e^{y \cdot 4}\right)}^{\left(\frac{y}{8}\right)}\right) \]
   pow-sqr [=>] 0.0	\[ x \cdot \color{blue}{{\left(e^{y \cdot 4}\right)}^{\left(2 \cdot \frac{y}{8}\right)}} \]

7. Final simplification 0.0

   \[\leadsto x \cdot {\left(e^{y \cdot 4}\right)}^{\left(2 \cdot \frac{y}{8}\right)} \]

Alternatives

Alternative 1
Error	0.0
Cost	13312

\[x \cdot {\left(e^{y + y}\right)}^{\left(y \cdot 0.5\right)} \]

Alternative 2
Error	0.0
Cost	13056

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

Alternative 3
Error	0.0
Cost	6720

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

Alternative 4
Error	0.5
Cost	1088

\[x + x \cdot \left(y \cdot y - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot -0.5\right) \]

Alternative 5
Error	0.6
Cost	448

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

Alternative 6
Error	0.6
Cost	448

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

Alternative 7
Error	0.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(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))))