Average Error: 0.7 → 0.3
Time: 6.4s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.9999999999132267:\\ \;\;\;\;\frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{e^{b} + 1} + \frac{a}{e^{b} + 1}\right) - \frac{a}{{\left(e^{b} + 1\right)}^{2}}\\ \end{array}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.9999999999132267:\\
\;\;\;\;\frac{e^{a}}{e^{a} + e^{b}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{e^{b} + 1} + \frac{a}{e^{b} + 1}\right) - \frac{a}{{\left(e^{b} + 1\right)}^{2}}\\

\end{array}
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.9999999999132267)
   (/ (exp a) (+ (exp a) (exp b)))
   (-
    (+ (/ 1.0 (+ (exp b) 1.0)) (/ a (+ (exp b) 1.0)))
    (/ a (pow (+ (exp b) 1.0) 2.0)))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.9999999999132267) {
		tmp = exp(a) / (exp(a) + exp(b));
	} else {
		tmp = ((1.0 / (exp(b) + 1.0)) + (a / (exp(b) + 1.0))) - (a / pow((exp(b) + 1.0), 2.0));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie0.3
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.99999999991322674

    1. Initial program 0.0

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

    if 0.99999999991322674 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

    1. Initial program 3.8

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\left(\frac{a}{e^{b} + 1} + \frac{1}{e^{b} + 1}\right) - \frac{a}{{\left(e^{b} + 1\right)}^{2}}}\]

    3. Simplified1.3

      \[\leadsto \color{blue}{\left(\frac{1}{e^{b} + 1} + \frac{a}{e^{b} + 1}\right) - \frac{a}{{\left(e^{b} + 1\right)}^{2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.9999999999132267:\\ \;\;\;\;\frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{e^{b} + 1} + \frac{a}{e^{b} + 1}\right) - \frac{a}{{\left(e^{b} + 1\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021007 
(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))))