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

Average Accuracy: 100.0% → 100.0%

Time: 4.7s

Precision: binary64

Cost: 32704

Math

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

↓

\[x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot e^{y \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (*
  x
  (* (pow (pow (cbrt (exp y)) 2.0) y) (exp (* y (* y 0.3333333333333333))))))

C

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

↓

double code(double x, double y) {
	return x * (pow(pow(cbrt(exp(y)), 2.0), y) * exp((y * (y * 0.3333333333333333))));
}

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.pow(Math.cbrt(Math.exp(y)), 2.0), y) * Math.exp((y * (y * 0.3333333333333333))));
}

Julia

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

↓

function code(x, y)
	return Float64(x * Float64(((cbrt(exp(y)) ^ 2.0) ^ y) * exp(Float64(y * Float64(y * 0.3333333333333333)))))
end

Wolfram

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

↓

code[x_, y_] := N[(x * N[(N[Power[N[Power[N[Power[N[Exp[y], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], y], $MachinePrecision] * N[Exp[N[(y * N[(y * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ x \cdot e^{y \cdot y} \]
   exp-prod [=>] 100.0	\[ x \cdot \color{blue}{{\left(e^{y}\right)}^{y}} \]

3. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{\left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\right)} \]
   Proof

   [Start] 100.0	\[ x \cdot {\left(e^{y}\right)}^{y} \]
   add-cube-cbrt [=>] 100.0	\[ x \cdot {\color{blue}{\left(\left(\sqrt[3]{e^{y}} \cdot \sqrt[3]{e^{y}}\right) \cdot \sqrt[3]{e^{y}}\right)}}^{y} \]
   unpow-prod-down [=>] 100.0	\[ x \cdot \color{blue}{\left({\left(\sqrt[3]{e^{y}} \cdot \sqrt[3]{e^{y}}\right)}^{y} \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\right)} \]
   pow2 [=>] 100.0	\[ x \cdot \left({\color{blue}{\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}}^{y} \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot \color{blue}{e^{y \cdot \left(y \cdot 0.3333333333333333\right)}}\right) \]
   Proof

   [Start] 100.0	\[ x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\right) \]
   add-exp-log [=>] 100.0	\[ x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot \color{blue}{e^{\log \left({\left(\sqrt[3]{e^{y}}\right)}^{y}\right)}}\right) \]
   log-pow [=>] 100.0	\[ x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot e^{\color{blue}{y \cdot \log \left(\sqrt[3]{e^{y}}\right)}}\right) \]
   pow1/3 [=>] 100.0	\[ x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot e^{y \cdot \log \color{blue}{\left({\left(e^{y}\right)}^{0.3333333333333333}\right)}}\right) \]
   pow-exp [=>] 100.0	\[ x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot e^{y \cdot \log \color{blue}{\left(e^{y \cdot 0.3333333333333333}\right)}}\right) \]
   add-log-exp [<=] 100.0	\[ x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot e^{y \cdot \color{blue}{\left(y \cdot 0.3333333333333333\right)}}\right) \]

5. Final simplification 100.0%

   \[\leadsto x \cdot \left({\left({\left(\sqrt[3]{e^{y}}\right)}^{2}\right)}^{y} \cdot e^{y \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	26240

\[x \cdot \left(e^{y \cdot \left(y \cdot 0.6666666666666666\right)} \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\right) \]

Alternative 2
Accuracy	100.0%
Cost	13056

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

Alternative 3
Accuracy	100.0%
Cost	6720

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

Alternative 4
Accuracy	99.0%
Cost	448

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

Alternative 5
Accuracy	99.0%
Cost	448

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

Alternative 6
Accuracy	98.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(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))))