symmetry log of sum of exp

Average Error: 45.97% → 1.56%

Time: 16.9s

Precision: binary64

Cost: 19648

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

Math

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

↓

\[\mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1} \]

FPCore

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

↓

(FPCore (a b) :precision binary64 (+ (log1p (exp a)) (/ b (+ (exp a) 1.0))))

C

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

↓

double code(double a, double b) {
	return log1p(exp(a)) + (b / (exp(a) + 1.0));
}

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)) + (b / (Math.exp(a) + 1.0));
}

Python

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

↓

def code(a, b):
	return math.log1p(math.exp(a)) + (b / (math.exp(a) + 1.0))

Julia

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

↓

function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / Float64(exp(a) + 1.0)))
end

Wolfram

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

↓

code[a_, b_] := N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.97

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

2. Taylor expanded in b around 0 1.77

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}}} \]

3. Simplified 1.56

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]
   Proof

   [Start] 1.77	\[ \log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}} \]
   log1p-def [=>] 1.56	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} + \frac{b}{1 + e^{a}} \]

4. Final simplification 1.56

   \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1} \]

Alternatives

Alternative 1
Error	1.73%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	2.29%
Cost	19396

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

Alternative 3
Error	2.1%
Cost	19392

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

Alternative 4
Error	1.87%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;a \leq -37:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \]

Alternative 5
Error	42.82%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 6
Error	2.92%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -37:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 7
Error	42.95%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + 2\right)\\ \end{array} \]

Alternative 8
Error	42.91%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -145:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 9
Error	43.45%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -160:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 10
Error	97.38%
Cost	192

\[a \cdot 0.5 \]

Alternative 11
Error	88.09%
Cost	192

\[b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023089 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))