Quotient of sum of exps

Average Error: 0.7 → 1.1

Time: 6.7s

Precision: binary64

Cost: 32456

Math

\[\frac{e^{a}}{e^{a} + e^{b}} \]

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999995:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 1.0000000001:\\ \;\;\;\;e^{a - \mathsf{log1p}\left(e^{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.9999995)
   (/ 1.0 (+ (exp b) 1.0))
   (if (<= (exp b) 1.0000000001)
     (exp (- a (log1p (exp a))))
     (exp (- (log1p (exp b)))))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 0.9999995) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else if (exp(b) <= 1.0000000001) {
		tmp = exp((a - log1p(exp(a))));
	} else {
		tmp = exp(-log1p(exp(b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

↓

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 0.9999995) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else if (Math.exp(b) <= 1.0000000001) {
		tmp = Math.exp((a - Math.log1p(Math.exp(a))));
	} else {
		tmp = Math.exp(-Math.log1p(Math.exp(b)));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if math.exp(b) <= 0.9999995:
		tmp = 1.0 / (math.exp(b) + 1.0)
	elif math.exp(b) <= 1.0000000001:
		tmp = math.exp((a - math.log1p(math.exp(a))))
	else:
		tmp = math.exp(-math.log1p(math.exp(b)))
	return tmp

Julia

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

↓

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 0.9999995)
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	elseif (exp(b) <= 1.0000000001)
		tmp = exp(Float64(a - log1p(exp(a))));
	else
		tmp = exp(Float64(-log1p(exp(b))));
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 0.9999995], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[b], $MachinePrecision], 1.0000000001], N[Exp[N[(a - N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]]]

Target

Original	0.7
Target	0.0
Herbie	1.1

\[\frac{1}{1 + e^{b - a}} \]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 0.999999500000000041

   1. Initial program 1.8

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 1.7

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

   if 0.999999500000000041 < (exp.f64 b) < 1.0000000001

   1. Initial program 0.5

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Applied egg-rr 0.4

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

   3. Taylor expanded in b around 0 0.7

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

   4. Simplified 0.7

      \[\leadsto e^{a - \color{blue}{\mathsf{log1p}\left(e^{a}\right)}} \]
      Proof
      (log1p.f64 (exp.f64 a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (exp.f64 a)))): 3 points increase in error, 3 points decrease in error

   if 1.0000000001 < (exp.f64 b)

   1. Initial program 0.6

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Applied egg-rr 0.1

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

   3. Taylor expanded in a around 0 1.4

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

   4. Simplified 1.4

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
      Proof
      (neg.f64 (log1p.f64 (exp.f64 b))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (exp.f64 b))))): 1 points increase in error, 2 points decrease in error
      (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (log.f64 (+.f64 1 (exp.f64 b))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999995:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 1.0000000001:\\ \;\;\;\;e^{a - \mathsf{log1p}\left(e^{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	32392

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999995:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 1.0000000001:\\ \;\;\;\;\frac{e^{a}}{1 + \left(e^{a} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]

Alternative 2
Error	1.1
Cost	26312

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.9999995:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.0000000001:\\ \;\;\;\;\frac{1}{\frac{e^{a} + 1}{e^{a}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.9
Cost	26312

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.9999995:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.0000000001:\\ \;\;\;\;\frac{e^{a}}{1 + \left(e^{a} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	26184

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.9999995:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.0000000001:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.6
Cost	25920

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

Alternative 6
Error	1.3
Cost	19912

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.9999995:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.0000000001:\\ \;\;\;\;\frac{e^{a}}{2 + \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.7
Cost	19520

\[\frac{e^{a}}{e^{a} + e^{b}} \]

Alternative 8
Error	1.0
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 9
Error	13.7
Cost	6860

\[\begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-241}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-278}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 10
Error	13.3
Cost	708

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

Alternative 11
Error	20.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 12
Error	29.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]

Alternative 13
Error	29.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 14
Error	29.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 15
Error	39.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022332 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))