Quotient of sum of exps

Average Error: 0.7 → 1.0

Time: 6.9s

Precision: binary64

Cost: 32392

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 10^{-27}:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;e^{b} \leq 1.00000000005:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 1e-27)
   (+ (exp b) 1.0)
   (if (<= (exp b) 1.00000000005)
     (/ (exp a) (+ (exp a) 1.0))
     (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) <= 1e-27) {
		tmp = exp(b) + 1.0;
	} else if (exp(b) <= 1.00000000005) {
		tmp = exp(a) / (exp(a) + 1.0);
	} 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) <= 1e-27) {
		tmp = Math.exp(b) + 1.0;
	} else if (Math.exp(b) <= 1.00000000005) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} 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) <= 1e-27:
		tmp = math.exp(b) + 1.0
	elif math.exp(b) <= 1.00000000005:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	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) <= 1e-27)
		tmp = Float64(exp(b) + 1.0);
	elseif (exp(b) <= 1.00000000005)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	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], 1e-27], N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[N[Exp[b], $MachinePrecision], 1.00000000005], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[(-N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]]]

Target

Original	0.7
Target	0.0
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 1e-27

   1. Initial program 2.0

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

   2. Applied egg-rr 2.0

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

   3. Taylor expanded in a around 0 0.9

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

   4. Simplified 0.9

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

      [Start] 0.9	\[ e^{-1 \cdot \log \left(1 + e^{b}\right)} \]
      mul-1-neg [=>] 0.9	\[ e^{\color{blue}{-\log \left(1 + e^{b}\right)}} \]
      log1p-def [=>] 0.9	\[ e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   5. Applied egg-rr 0.9

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

   if 1e-27 < (exp.f64 b) < 1.00000000005

   1. Initial program 0.4

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

   2. Taylor expanded in b around 0 1.0

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

   if 1.00000000005 < (exp.f64 b)

   1. Initial program 0.2

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

   2. Applied egg-rr 0.0

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

   3. Taylor expanded in a around 0 1.0

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

   4. Simplified 1.0

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

      [Start] 1.0	\[ e^{-1 \cdot \log \left(1 + e^{b}\right)} \]
      mul-1-neg [=>] 1.0	\[ e^{\color{blue}{-\log \left(1 + e^{b}\right)}} \]
      log1p-def [=>] 1.0	\[ e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 10^{-27}:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;e^{b} \leq 1.00000000005:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	26184

\[\begin{array}{l} t_0 := e^{b} + 1\\ \mathbf{if}\;e^{b} \leq 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.00000000005:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	25920

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

Alternative 3
Error	0.7
Cost	19520

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

Alternative 4
Error	15.4
Cost	7124

\[\begin{array}{l} t_0 := 0.5 + a \cdot 0.25\\ \mathbf{if}\;b \leq -0.28:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -2.15 \cdot 10^{-192}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-207}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{1}{b + 2}\right)\\ \end{array} \]

Alternative 5
Error	15.0
Cost	7124

\[\begin{array}{l} t_0 := 0.5 + a \cdot 0.25\\ \mathbf{if}\;b \leq -0.29:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-192}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-207}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{1}{b + 2}\right)\\ \end{array} \]

Alternative 6
Error	1.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -7300000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 7
Error	24.0
Cost	980

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-132}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-273}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-207}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.032:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	25.1
Cost	980

\[\begin{array}{l} t_0 := 0.5 + a \cdot 0.25\\ \mathbf{if}\;b \leq -3.6 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.72 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-207}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.112:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	24.4
Cost	724

\[\begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-131}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 3.55 \cdot 10^{-273}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-207}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	13.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{1}{b + 2}\right)\\ \end{array} \]

Alternative 11
Error	39.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023032 
(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))))