Quotient of sum of exps

Average Error: 0.5 → 0.7

Time: 6.2s

Precision: binary64

Cost: 19849

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99999999998 \lor \neg \left(e^{b} \leq 1.2\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.99999999998) (not (<= (exp b) 1.2)))
   (/ 1.0 (+ (exp b) 1.0))
   (/ 1.0 (+ 1.0 (exp (- a))))))

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) <= 0.99999999998) || !(exp(b) <= 1.2)) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(-a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((exp(b) <= 0.99999999998d0) .or. (.not. (exp(b) <= 1.2d0))) then
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    end if
    code = tmp
end function

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) <= 0.99999999998) || !(Math.exp(b) <= 1.2)) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	}
	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) <= 0.99999999998) or not (math.exp(b) <= 1.2):
		tmp = 1.0 / (math.exp(b) + 1.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(-a))
	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) <= 0.99999999998) || !(exp(b) <= 1.2))
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(b) <= 0.99999999998) || ~((exp(b) <= 1.2)))
		tmp = 1.0 / (exp(b) + 1.0);
	else
		tmp = 1.0 / (1.0 + exp(-a));
	end
	tmp_2 = 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[Or[LessEqual[N[Exp[b], $MachinePrecision], 0.99999999998], N[Not[LessEqual[N[Exp[b], $MachinePrecision], 1.2]], $MachinePrecision]], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.5
Target	0.0
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.99999999998 or 1.19999999999999996 < (exp.f64 b)

   1. Initial program 0.8

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

   2. Taylor expanded in a around 0 1.1

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

   if 0.99999999998 < (exp.f64 b) < 1.19999999999999996

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

   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

      [Start] 0.7	\[ e^{a - \log \left(1 + e^{a}\right)} \]
      log1p-def [=>] 0.7	\[ e^{a - \color{blue}{\mathsf{log1p}\left(e^{a}\right)}} \]

   5. Applied egg-rr 0.7

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

   6. Taylor expanded in a around inf 0.7

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

   7. Simplified 0.4

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

      [Start] 0.7	\[ \frac{1}{\frac{1 + e^{a}}{e^{a}}} \]
      +-commutative [<=] 0.7	\[ \frac{1}{\frac{\color{blue}{e^{a} + 1}}{e^{a}}} \]
      *-lft-identity [<=] 0.7	\[ \frac{1}{\frac{\color{blue}{1 \cdot \left(e^{a} + 1\right)}}{e^{a}}} \]
      associate-*l/ [<=] 0.7	\[ \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot \left(e^{a} + 1\right)}} \]
      exp-neg [<=] 0.7	\[ \frac{1}{\color{blue}{e^{-a}} \cdot \left(e^{a} + 1\right)} \]
      distribute-rgt-in [=>] 21.6	\[ \frac{1}{\color{blue}{e^{a} \cdot e^{-a} + 1 \cdot e^{-a}}} \]
      *-lft-identity [=>] 21.6	\[ \frac{1}{e^{a} \cdot e^{-a} + \color{blue}{e^{-a}}} \]
      exp-neg [=>] 21.6	\[ \frac{1}{e^{a} \cdot \color{blue}{\frac{1}{e^{a}}} + e^{-a}} \]
      rgt-mult-inverse [=>] 0.4	\[ \frac{1}{\color{blue}{1} + e^{-a}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99999999998 \lor \neg \left(e^{b} \leq 1.2\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	25920

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

Alternative 2
Error	0.5
Cost	19520

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

Alternative 3
Error	0.8
Cost	13252

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

Alternative 4
Error	23.5
Cost	7124

\[\begin{array}{l} t_0 := -0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-258}:\\ \;\;\;\;t_0 + \left(0.5 + a \cdot 0.25\right)\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-290}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+180}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	30.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 550:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 6
Error	38.6
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 7
Error	38.2
Cost	320

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

Alternative 8
Error	38.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022354 
(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))))