Quotient of sum of exps

Average Accuracy: 99.0% → 99.0%

Time: 6.4s

Precision: binary64

Cost: 25860

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.99)
   (exp (- (log1p (exp b))))
   (if (<= (exp b) 2.0) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ (exp b) 1.0)))))

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) <= 0.99) {
		tmp = exp(-log1p(exp(b)));
	} else if (exp(b) <= 2.0) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	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) <= 0.99) {
		tmp = Math.exp(-Math.log1p(Math.exp(b)));
	} else if (Math.exp(b) <= 2.0) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	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) <= 0.99:
		tmp = math.exp(-math.log1p(math.exp(b)))
	elif math.exp(b) <= 2.0:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	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) <= 0.99)
		tmp = exp(Float64(-log1p(exp(b))));
	elseif (exp(b) <= 2.0)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	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], 0.99], N[Exp[(-N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision])], $MachinePrecision], If[LessEqual[N[Exp[b], $MachinePrecision], 2.0], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	99.0%
Target	100.0%
Herbie	99.0%

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 0.98999999999999999

   1. Initial program 97.2%

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

   2. Applied egg-rr 97.2%

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

      [Start] 97.2	\[ \frac{e^{a}}{e^{a} + e^{b}} \]
      add-exp-log [=>] 97.2	\[ \frac{e^{a}}{\color{blue}{e^{\log \left(e^{a} + e^{b}\right)}}} \]
      div-exp [=>] 97.2	\[ \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}} \]

   3. Taylor expanded in a around 0 98.8%

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

   4. Simplified 98.8%

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

      [Start] 98.8	\[ e^{-1 \cdot \log \left(1 + e^{b}\right)} \]
      mul-1-neg [=>] 98.8	\[ e^{\color{blue}{-\log \left(1 + e^{b}\right)}} \]
      log1p-def [=>] 98.8	\[ e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   if 0.98999999999999999 < (exp.f64 b) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in b around 0 98.2%

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

   3. Applied egg-rr 98.1%

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

      [Start] 98.2	\[ \frac{e^{a}}{1 + e^{a}} \]
      frac-2neg [=>] 98.2	\[ \color{blue}{\frac{-e^{a}}{-\left(1 + e^{a}\right)}} \]
      div-inv [=>] 98.1	\[ \color{blue}{\left(-e^{a}\right) \cdot \frac{1}{-\left(1 + e^{a}\right)}} \]
      +-commutative [=>] 98.1	\[ \left(-e^{a}\right) \cdot \frac{1}{-\color{blue}{\left(e^{a} + 1\right)}} \]
      distribute-neg-in [=>] 98.1	\[ \left(-e^{a}\right) \cdot \frac{1}{\color{blue}{\left(-e^{a}\right) + \left(-1\right)}} \]
      metadata-eval [=>] 98.1	\[ \left(-e^{a}\right) \cdot \frac{1}{\left(-e^{a}\right) + \color{blue}{-1}} \]

   4. Simplified 98.7%

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

      [Start] 98.1	\[ \left(-e^{a}\right) \cdot \frac{1}{\left(-e^{a}\right) + -1} \]
      associate-*r/ [=>] 98.2	\[ \color{blue}{\frac{\left(-e^{a}\right) \cdot 1}{\left(-e^{a}\right) + -1}} \]
      *-rgt-identity [=>] 98.2	\[ \frac{\color{blue}{-e^{a}}}{\left(-e^{a}\right) + -1} \]
      neg-mul-1 [=>] 98.2	\[ \frac{\color{blue}{-1 \cdot e^{a}}}{\left(-e^{a}\right) + -1} \]
      associate-/l* [=>] 98.1	\[ \color{blue}{\frac{-1}{\frac{\left(-e^{a}\right) + -1}{e^{a}}}} \]
      +-commutative [=>] 98.1	\[ \frac{-1}{\frac{\color{blue}{-1 + \left(-e^{a}\right)}}{e^{a}}} \]
      unsub-neg [=>] 98.1	\[ \frac{-1}{\frac{\color{blue}{-1 - e^{a}}}{e^{a}}} \]
      div-sub [=>] 63.8	\[ \frac{-1}{\color{blue}{\frac{-1}{e^{a}} - \frac{e^{a}}{e^{a}}}} \]
      *-inverses [=>] 98.7	\[ \frac{-1}{\frac{-1}{e^{a}} - \color{blue}{1}} \]
      sub-neg [=>] 98.7	\[ \frac{-1}{\color{blue}{\frac{-1}{e^{a}} + \left(-1\right)}} \]
      metadata-eval [=>] 98.7	\[ \frac{-1}{\frac{-1}{e^{a}} + \color{blue}{-1}} \]

   5. Taylor expanded in a around inf 98.7%

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

   6. Simplified 98.7%

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

      [Start] 98.7	\[ \frac{1}{1 + \frac{1}{e^{a}}} \]
      metadata-eval [<=] 98.7	\[ \frac{1}{1 + \frac{\color{blue}{--1}}{e^{a}}} \]
      distribute-neg-frac [<=] 98.7	\[ \frac{1}{1 + \color{blue}{\left(-\frac{-1}{e^{a}}\right)}} \]
      distribute-neg-frac [=>] 98.7	\[ \frac{1}{1 + \color{blue}{\frac{--1}{e^{a}}}} \]
      metadata-eval [=>] 98.7	\[ \frac{1}{1 + \frac{\color{blue}{1}}{e^{a}}} \]
      exp-neg [<=] 98.7	\[ \frac{1}{1 + \color{blue}{e^{-a}}} \]

   if 2 < (exp.f64 b)

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 99.7%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	25920

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

Alternative 2
Accuracy	99.0%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99 \lor \neg \left(e^{b} \leq 2\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 3
Accuracy	99.0%
Cost	19520

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

Alternative 4
Accuracy	78.7%
Cost	13124

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

Alternative 5
Accuracy	97.8%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+15}:\\ \;\;\;\;-1 + \left(1 + \frac{-1}{a + -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6
Accuracy	77.9%
Cost	6596

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

Alternative 7
Accuracy	63.8%
Cost	576

\[-1 + \left(1 + \frac{-1}{a + -2}\right) \]

Alternative 8
Accuracy	52.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq -1.7:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 9
Accuracy	38.4%
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 10
Accuracy	39.0%
Cost	320

\[\frac{-1}{a + -2} \]

Alternative 11
Accuracy	38.2%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023138 
(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))))