symmetry log of sum of exp

Average Error: 29.6 → 0.8

Time: 17.2s

Precision: binary64

Cost: 13128

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -390:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -390.0)
   (/ b (+ 2.0 (expm1 a)))
   (if (<= a -5e-45) (log1p (exp a)) (log1p (exp b)))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (a <= -390.0) {
		tmp = b / (2.0 + expm1(a));
	} else if (a <= -5e-45) {
		tmp = log1p(exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

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) {
	double tmp;
	if (a <= -390.0) {
		tmp = b / (2.0 + Math.expm1(a));
	} else if (a <= -5e-45) {
		tmp = Math.log1p(Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if a <= -390.0:
		tmp = b / (2.0 + math.expm1(a))
	elif a <= -5e-45:
		tmp = math.log1p(math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

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

↓

function code(a, b)
	tmp = 0.0
	if (a <= -390.0)
		tmp = Float64(b / Float64(2.0 + expm1(a)));
	elseif (a <= -5e-45)
		tmp = log1p(exp(a));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_] := If[LessEqual[a, -390.0], N[(b / N[(2.0 + N[(Exp[a] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -5e-45], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -390

   1. Initial program 58.3

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

   2. Taylor expanded in b around 0 0.0

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

   3. Simplified 0

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

      [Start] 0.0	\[ \log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}} \]
      log1p-def [=>] 0	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} + \frac{b}{1 + e^{a}} \]

   4. Taylor expanded in b around inf 0.0

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   5. Simplified 0.0

      \[\leadsto \color{blue}{\frac{b}{2 + \mathsf{expm1}\left(a\right)}} \]
      Proof

      [Start] 0.0	\[ \frac{b}{1 + e^{a}} \]
      +-commutative [=>] 0.0	\[ \frac{b}{\color{blue}{e^{a} + 1}} \]
      metadata-eval [<=] 0.0	\[ \frac{b}{e^{a} + \color{blue}{\left(2 - 1\right)}} \]
      associate--l+ [<=] 0.0	\[ \frac{b}{\color{blue}{\left(e^{a} + 2\right) - 1}} \]
      +-commutative [<=] 0.0	\[ \frac{b}{\color{blue}{\left(2 + e^{a}\right)} - 1} \]
      associate--l+ [=>] 0.0	\[ \frac{b}{\color{blue}{2 + \left(e^{a} - 1\right)}} \]
      expm1-def [=>] 0.0	\[ \frac{b}{2 + \color{blue}{\mathsf{expm1}\left(a\right)}} \]

   if -390 < a < -4.99999999999999976e-45

   1. Initial program 2.4

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

   2. Taylor expanded in b around 0 3.5

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

   3. Simplified 1.9

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

      [Start] 3.5	\[ \log \left(1 + e^{a}\right) \]
      log1p-def [=>] 1.9	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

   if -4.99999999999999976e-45 < a

   1. Initial program 1.5

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

   2. Taylor expanded in a around 0 1.5

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

   3. Simplified 1.5

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

      [Start] 1.5	\[ \log \left(1 + e^{b}\right) \]
      log1p-def [=>] 1.5	\[ \color{blue}{\mathsf{log1p}\left(e^{b}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -390:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	25924

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

Alternative 2
Error	1.2
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 3
Error	1.3
Cost	19652

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

Alternative 4
Error	1.5
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 5
Error	1.4
Cost	19392

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

Alternative 6
Error	1.7
Cost	7108

\[\begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b + 2} + \log \left(b + 2\right)\\ \end{array} \]

Alternative 7
Error	27.4
Cost	6852

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

Alternative 8
Error	1.7
Cost	6852

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

Alternative 9
Error	27.7
Cost	6724

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

Alternative 10
Error	28.0
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -170:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 11
Error	56.3
Cost	192

\[\frac{b}{2} \]

Reproduce

herbie shell --seed 2023073 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))