Average Error: 0.7 → 0.4
Time: 3.3s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 3.421041865160871 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\begin{array}{l}
\mathbf{if}\;e^{a} \leq 3.421041865160871 \cdot 10^{+102}:\\
\;\;\;\;\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 + e^{b}}\\

\end{array}
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 3.421041865160871e+102)
   (/ 1.0 (/ (+ (exp a) (exp b)) (exp a)))
   (/ 1.0 (+ 1.0 (exp b)))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 3.421041865160871e+102) {
		tmp = 1.0 / ((exp(a) + exp(b)) / exp(a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	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
Herbie0.4
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp.f64 a) < 3.42104186516087124e102

    1. Initial program 0.2

      \[\frac{e^{a}}{e^{a} + e^{b}}\]
    2. Using strategy rm

    3. Applied clear-num_binary64_24640.3

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

    4. Simplified0.3

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

    if 3.42104186516087124e102 < (exp.f64 a)

    1. Initial program 60.7

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

    2. Taylor expanded around 0 25.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 3.421041865160871 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))