Quotient of sum of exps

Average Error: 0.8 → 1.0

Time: 6.9s

Precision: binary64

Cost: 13385

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -63000 \lor \neg \left(b \leq 1.45 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (or (<= b -63000.0) (not (<= b 1.45e-11)))
   (/ 1.0 (+ (exp b) 1.0))
   (/ (exp a) (+ (exp a) 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 ((b <= -63000.0) || !(b <= 1.45e-11)) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else {
		tmp = exp(a) / (exp(a) + 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 ((b <= (-63000.0d0)) .or. (.not. (b <= 1.45d-11))) then
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    else
        tmp = exp(a) / (exp(a) + 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 ((b <= -63000.0) || !(b <= 1.45e-11)) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else {
		tmp = Math.exp(a) / (Math.exp(a) + 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 (b <= -63000.0) or not (b <= 1.45e-11):
		tmp = 1.0 / (math.exp(b) + 1.0)
	else:
		tmp = math.exp(a) / (math.exp(a) + 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 ((b <= -63000.0) || !(b <= 1.45e-11))
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	else
		tmp = Float64(exp(a) / Float64(exp(a) + 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 ((b <= -63000.0) || ~((b <= 1.45e-11)))
		tmp = 1.0 / (exp(b) + 1.0);
	else
		tmp = exp(a) / (exp(a) + 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[b, -63000.0], N[Not[LessEqual[b, 1.45e-11]], $MachinePrecision]], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.8
Target	0.0
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if b < -63000 or 1.45e-11 < b

   1. Initial program 0.9

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

   2. Taylor expanded in a around 0 0.7

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

   if -63000 < b < 1.45e-11

   1. Initial program 0.6

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

   2. Taylor expanded in b around 0 1.3

      \[\leadsto \color{blue}{\frac{e^{a}}{1 + e^{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -63000 \lor \neg \left(b \leq 1.45 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	25920

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

Alternative 2
Error	0.8
Cost	19520

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

Alternative 3
Error	1.1
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4
Error	13.4
Cost	12996

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 5
Error	23.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-245}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 6
Error	30.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.35:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 7
Error	38.8
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 8
Error	39.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023083 
(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))))