symmetry log of sum of exp

Average Error: 46.38% → 2.2%

Time: 14.4s

Precision: binary64

Cost: 19392

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]

Math

\[\log \left(e^{a} + e^{b}\right) \]

↓

\[\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \]

FPCore

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))

↓

(FPCore (a b) :precision binary64 (log1p (+ (exp a) (expm1 b))))

C

double code(double a, double b) {
	return log((exp(a) + exp(b)));
}

↓

double code(double a, double b) {
	return log1p((exp(a) + expm1(b)));
}

Java

public static double code(double a, double b) {
	return Math.log((Math.exp(a) + Math.exp(b)));
}

↓

public static double code(double a, double b) {
	return Math.log1p((Math.exp(a) + Math.expm1(b)));
}

Python

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))

↓

def code(a, b):
	return math.log1p((math.exp(a) + math.expm1(b)))

Julia

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end

↓

function code(a, b)
	return log1p(Float64(exp(a) + expm1(b)))
end

Wolfram

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[a_, b_] := N[Log[1 + N[(N[Exp[a], $MachinePrecision] + N[(Exp[b] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 46.38

   \[\log \left(e^{a} + e^{b}\right) \]

2. Applied egg-rr 47.12

   \[\leadsto \color{blue}{\log \left(\sqrt{e^{a} + e^{b}}\right) + \log \left(\sqrt{e^{a} + e^{b}}\right)} \]

3. Simplified 2.2

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right)} \]
   Proof

   [Start] 47.12	\[ \log \left(\sqrt{e^{a} + e^{b}}\right) + \log \left(\sqrt{e^{a} + e^{b}}\right) \]
   log-prod [<=] 47.53	\[ \color{blue}{\log \left(\sqrt{e^{a} + e^{b}} \cdot \sqrt{e^{a} + e^{b}}\right)} \]
   rem-square-sqrt [=>] 46.38	\[ \log \color{blue}{\left(e^{a} + e^{b}\right)} \]
   log1p-expm1 [<=] 46.56	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{a} + e^{b}\right)\right)\right)} \]
   expm1-def [<=] 46.56	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(e^{a} + e^{b}\right)} - 1}\right) \]
   rem-exp-log [=>] 46.56	\[ \mathsf{log1p}\left(\color{blue}{\left(e^{a} + e^{b}\right)} - 1\right) \]
   associate--l+ [=>] 46.29	\[ \mathsf{log1p}\left(\color{blue}{e^{a} + \left(e^{b} - 1\right)}\right) \]
   expm1-def [=>] 2.2	\[ \mathsf{log1p}\left(e^{a} + \color{blue}{\mathsf{expm1}\left(b\right)}\right) \]

4. Final simplification 2.2

   \[\leadsto \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \]

Alternatives

Alternative 1
Error	3.21%
Cost	25924

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 1.00000002:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternative 2
Error	47.84%
Cost	13128

\[\begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{elif}\;b \leq 720:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 3
Error	4.28%
Cost	12992

\[\mathsf{log1p}\left(e^{a} + b\right) \]

Alternative 4
Error	49.52%
Cost	12864

\[\mathsf{log1p}\left(e^{a}\right) \]

Alternative 5
Error	50.88%
Cost	6976

\[\log \left(2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right) \]

Alternative 6
Error	50.78%
Cost	6720

\[b \cdot 0.5 + \log 2 \]

Alternative 7
Error	51.06%
Cost	6592

\[\log \left(b + 2\right) \]

Alternative 8
Error	51.05%
Cost	6592

\[\mathsf{log1p}\left(b + 1\right) \]

Alternative 9
Error	51.56%
Cost	6464

\[\log 2 \]

Alternative 10
Error	97.37%
Cost	192

\[a \cdot 0.5 \]

Reproduce

herbie shell --seed 2023103 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))