symmetry log of sum of exp

Average Error: 29.3 → 1.2

Time: 17.2s

Precision: binary64

Cost: 45513

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{a} + e^{b}\\ \mathbf{if}\;t_0 \leq 1.5 \lor \neg \left(t_0 \leq 2.1\right):\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) (exp b))))
   (if (or (<= t_0 1.5) (not (<= t_0 2.1)))
     (/ b (+ 2.0 (expm1 a)))
     (log t_0))))

C

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

↓

double code(double a, double b) {
	double t_0 = exp(a) + exp(b);
	double tmp;
	if ((t_0 <= 1.5) || !(t_0 <= 2.1)) {
		tmp = b / (2.0 + expm1(a));
	} else {
		tmp = log(t_0);
	}
	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 t_0 = Math.exp(a) + Math.exp(b);
	double tmp;
	if ((t_0 <= 1.5) || !(t_0 <= 2.1)) {
		tmp = b / (2.0 + Math.expm1(a));
	} else {
		tmp = Math.log(t_0);
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	t_0 = math.exp(a) + math.exp(b)
	tmp = 0
	if (t_0 <= 1.5) or not (t_0 <= 2.1):
		tmp = b / (2.0 + math.expm1(a))
	else:
		tmp = math.log(t_0)
	return tmp

Julia

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

↓

function code(a, b)
	t_0 = Float64(exp(a) + exp(b))
	tmp = 0.0
	if ((t_0 <= 1.5) || !(t_0 <= 2.1))
		tmp = Float64(b / Float64(2.0 + expm1(a)));
	else
		tmp = log(t_0);
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 1.5], N[Not[LessEqual[t$95$0, 2.1]], $MachinePrecision]], N[(b / N[(2.0 + N[(Exp[a] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.5 or 2.10000000000000009 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 57.6

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

   2. Taylor expanded in b around 0 1.9

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

   3. Simplified 1.7

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

      [Start] 1.9	\[ \log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}} \]
      log1p-def [=>] 1.7	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} + \frac{b}{1 + e^{a}} \]

   4. Taylor expanded in b around inf 2.3

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

   5. Simplified 2.3

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

      [Start] 2.3	\[ \frac{b}{1 + e^{a}} \]
      +-commutative [=>] 2.3	\[ \frac{b}{\color{blue}{e^{a} + 1}} \]
      metadata-eval [<=] 2.3	\[ \frac{b}{e^{a} + \color{blue}{\left(2 - 1\right)}} \]
      associate--l+ [<=] 2.3	\[ \frac{b}{\color{blue}{\left(e^{a} + 2\right) - 1}} \]
      +-commutative [<=] 2.3	\[ \frac{b}{\color{blue}{\left(2 + e^{a}\right)} - 1} \]
      associate--l+ [=>] 2.3	\[ \frac{b}{\color{blue}{2 + \left(e^{a} - 1\right)}} \]
      expm1-def [=>] 2.3	\[ \frac{b}{2 + \color{blue}{\mathsf{expm1}\left(a\right)}} \]

   if 1.5 < (+.f64 (exp.f64 a) (exp.f64 b)) < 2.10000000000000009

   1. Initial program 0.0

      \[\log \left(e^{a} + e^{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.5 \lor \neg \left(e^{a} + e^{b} \leq 2.1\right):\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	25924

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

Alternative 2
Error	1.1
Cost	19648

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

Alternative 3
Error	1.7
Cost	19396

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-301}:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 4
Error	1.1
Cost	13636

\[\begin{array}{l} \mathbf{if}\;a \leq -36:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	1.8
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-301}:\\ \;\;\;\;\frac{b}{2 + \mathsf{expm1}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \]

Alternative 6
Error	27.1
Cost	6852

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

Alternative 7
Error	27.4
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 8
Error	27.4
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \]

Alternative 9
Error	27.7
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 10
Error	56.4
Cost	192

\[b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023083 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))