symmetry log of sum of exp

Average Error: 29.9 → 0.8

Time: 20.3s

Precision: binary64

Cost: 26052

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{a} + 1\\ \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{b}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\log \left(t_0 + \mathsf{expm1}\left(b\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (if (<= (exp a) 2e-19) (/ b t_0) (log (+ t_0 (expm1 b))))))

C

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

↓

double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	double tmp;
	if (exp(a) <= 2e-19) {
		tmp = b / t_0;
	} else {
		tmp = log((t_0 + expm1(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 t_0 = Math.exp(a) + 1.0;
	double tmp;
	if (Math.exp(a) <= 2e-19) {
		tmp = b / t_0;
	} else {
		tmp = Math.log((t_0 + Math.expm1(b)));
	}
	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) + 1.0
	tmp = 0
	if math.exp(a) <= 2e-19:
		tmp = b / t_0
	else:
		tmp = math.log((t_0 + math.expm1(b)))
	return tmp

Julia

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

↓

function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	tmp = 0.0
	if (exp(a) <= 2e-19)
		tmp = Float64(b / t_0);
	else
		tmp = log(Float64(t_0 + expm1(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_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 2e-19], N[(b / t$95$0), $MachinePrecision], N[Log[N[(t$95$0 + N[(Exp[b] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2e-19

   1. Initial program 58.3

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

   2. Taylor expanded in b around 0 0.2

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

   3. Simplified 0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]
      Proof
      (+.f64 (log1p.f64 (exp.f64 a)) (/.f64 b (+.f64 1 (exp.f64 a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (exp.f64 a)))) (/.f64 b (+.f64 1 (exp.f64 a)))): 4 points increase in error, 1 points decrease in error

   4. Taylor expanded in b around inf 0.2

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

   if 2e-19 < (exp.f64 a)

   1. Initial program 1.5

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

   2. Applied egg-rr 1.5

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

   3. Applied egg-rr 1.5

      \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + \mathsf{expm1}\left(b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(e^{a} + 1\right) + \mathsf{expm1}\left(b\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	83712

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ t_1 := \sqrt[3]{t_0}\\ t_0 + \frac{b}{{\left(e^{{\left({\left(\sqrt[3]{t_1}\right)}^{3}\right)}^{2}}\right)}^{t_1}} \end{array} \]

Alternative 2
Error	0.8
Cost	25924

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternative 3
Error	1.1
Cost	19648

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

Alternative 4
Error	1.6
Cost	19396

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

Alternative 5
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 6
Error	2.0
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 7
Error	32.5
Cost	6720

\[\log 2 + b \cdot 0.5 \]

Alternative 8
Error	32.7
Cost	6592

\[\log \left(b + 2\right) \]

Alternative 9
Error	33.0
Cost	6464

\[\log 2 \]

Alternative 10
Error	62.3
Cost	192

\[a \cdot 0.5 \]

Reproduce

herbie shell --seed 2022311 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))