Quotient of sum of exps

Average Error: 0.6 → 0.5

Time: 5.5s

Precision: binary64

Cost: 26313

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.995 \lor \neg \left(e^{b} \leq 2\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(b + 1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.995) (not (<= (exp b) 2.0)))
   (/ 1.0 (+ (exp b) 1.0))
   (/ (exp a) (+ (exp a) (+ 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) <= 0.995) || !(exp(b) <= 2.0)) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else {
		tmp = exp(a) / (exp(a) + (b + 1.0));
	}
	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) <= 0.995d0) .or. (.not. (exp(b) <= 2.0d0))) then
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    else
        tmp = exp(a) / (exp(a) + (b + 1.0d0))
    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) <= 0.995) || !(Math.exp(b) <= 2.0)) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else {
		tmp = Math.exp(a) / (Math.exp(a) + (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) <= 0.995) or not (math.exp(b) <= 2.0):
		tmp = 1.0 / (math.exp(b) + 1.0)
	else:
		tmp = math.exp(a) / (math.exp(a) + (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) <= 0.995) || !(exp(b) <= 2.0))
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	else
		tmp = Float64(exp(a) / Float64(exp(a) + Float64(b + 1.0)));
	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) <= 0.995) || ~((exp(b) <= 2.0)))
		tmp = 1.0 / (exp(b) + 1.0);
	else
		tmp = exp(a) / (exp(a) + (b + 1.0));
	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], 0.995], N[Not[LessEqual[N[Exp[b], $MachinePrecision], 2.0]], $MachinePrecision]], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.6
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.994999999999999996 or 2 < (exp.f64 b)

   1. Initial program 1.0

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

   2. Taylor expanded in a around 0 0.4

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

   if 0.994999999999999996 < (exp.f64 b) < 2

   1. Initial program 0.3

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

   2. Taylor expanded in b around 0 0.6

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

   3. Simplified 0.6

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

      [Start] 0.6	\[ \frac{e^{a}}{1 + \left(e^{a} + b\right)} \]
      associate-+r+ [=>] 0.6	\[ \frac{e^{a}}{\color{blue}{\left(1 + e^{a}\right) + b}} \]
      +-commutative [=>] 0.6	\[ \frac{e^{a}}{\color{blue}{\left(e^{a} + 1\right)} + b} \]
      associate-+l+ [=>] 0.6	\[ \frac{e^{a}}{\color{blue}{e^{a} + \left(1 + b\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.995 \lor \neg \left(e^{b} \leq 2\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(b + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	26185

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.995 \lor \neg \left(e^{b} \leq 2\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	19520

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

Alternative 3
Error	0.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4
Error	11.2
Cost	6724

\[\begin{array}{l} \mathbf{if}\;b \leq 350:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	13.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -340:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 6
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 7
Error	22.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-8}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	38.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023027 
(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))))