symmetry log of sum of exp

Average Error: 30.3 → 0.7

Time: 17.0s

Precision: binary64

Cost: 25928

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999999999996:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (if (<= (exp a) 0.99999999999996) (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 (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else if (exp(a) <= 0.99999999999996) {
		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 (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else if (Math.exp(a) <= 0.99999999999996) {
		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 math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	elif math.exp(a) <= 0.99999999999996:
		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 (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	elseif (exp(a) <= 0.99999999999996)
		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[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999999999996], 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 (exp.f64 a) < 0.0

   1. Initial program 58.4

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

   2. Taylor expanded in b around 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	\[ \log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}} \]
      log1p-def [=>] 0	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} + \frac{b}{1 + e^{a}} \]

   4. Taylor expanded in b around inf 0

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

   if 0.0 < (exp.f64 a) < 0.99999999999996003

   1. Initial program 6.6

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

   2. Taylor expanded in b around 0 7.8

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

   3. Simplified 3.4

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

      [Start] 7.8	\[ \log \left(1 + e^{a}\right) \]
      log1p-def [=>] 3.4	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

   if 0.99999999999996003 < (exp.f64 a)

   1. Initial program 1.4

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

   2. Taylor expanded in a around 0 1.4

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

   3. Simplified 1.4

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

      [Start] 1.4	\[ \log \left(1 + e^{b}\right) \]
      log1p-def [=>] 1.4	\[ \color{blue}{\mathsf{log1p}\left(e^{b}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999999999996:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	25924

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-61}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternative 2
Error	1.9
Cost	19908

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 1.005:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]

Alternative 3
Error	1.5
Cost	19396

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

Alternative 4
Error	1.4
Cost	19392

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

Alternative 5
Error	1.6
Cost	13764

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-61}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(0.5 + b \cdot -0.25\right) + \left(b \cdot 0.5 + \log 2\right)\\ \end{array} \]

Alternative 6
Error	1.6
Cost	13636

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-61}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(b + 0.5 \cdot \left(b \cdot b\right)\right) + \left(a + 2\right)\right)\\ \end{array} \]

Alternative 7
Error	2.0
Cost	13508

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-61}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(0.5 + b \cdot -0.25\right) + \log 2\\ \end{array} \]

Alternative 8
Error	28.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -370:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 9
Error	1.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -370:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 10
Error	28.2
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -370:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 11
Error	28.5
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -370:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 12
Error	56.2
Cost	192

\[b \cdot 0.5 \]

Reproduce

herbie shell --seed 2022356 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))