symmetry log of sum of exp

Average Error: 29.4 → 1.4

Time: 15.7s

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 29.4

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

2. Applied egg-rr 29.9

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

3. Simplified 1.4

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

   [Start] 29.9	\[ \log \left(\sqrt{e^{a} + e^{b}}\right) + \log \left(\sqrt{e^{a} + e^{b}}\right) \]
   log-prod [<=] 30.2	\[ \color{blue}{\log \left(\sqrt{e^{a} + e^{b}} \cdot \sqrt{e^{a} + e^{b}}\right)} \]
   rem-square-sqrt [=>] 29.4	\[ \log \color{blue}{\left(e^{a} + e^{b}\right)} \]
   log1p-expm1 [<=] 29.5	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{a} + e^{b}\right)\right)\right)} \]
   expm1-def [<=] 29.5	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(e^{a} + e^{b}\right)} - 1}\right) \]
   rem-exp-log [=>] 29.5	\[ \mathsf{log1p}\left(\color{blue}{\left(e^{a} + e^{b}\right)} - 1\right) \]
   associate--l+ [=>] 29.3	\[ \mathsf{log1p}\left(\color{blue}{e^{a} + \left(e^{b} - 1\right)}\right) \]
   expm1-def [=>] 1.4	\[ \mathsf{log1p}\left(e^{a} + \color{blue}{\mathsf{expm1}\left(b\right)}\right) \]

4. Final simplification 1.4

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

Alternatives

Alternative 1
Error	2.1
Cost	25928

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 1.001:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \mathbf{elif}\;e^{b} \leq 10^{+280}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 2
Error	1.8
Cost	25924

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 1.00000000001:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternative 3
Error	27.0
Cost	19396

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

Alternative 4
Error	27.3
Cost	19396

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 5
Error	2.2
Cost	13376

\[\mathsf{log1p}\left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right) \]

Alternative 6
Error	27.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -1.38:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]

Alternative 7
Error	27.6
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -380:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 8
Error	27.9
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -380:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 9
Error	56.4
Cost	192

\[b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023028 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))