Average Error: 0.7 → 0.8
Time: 3.0s
Precision: binary64
Cost: 7041
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -2277.436057145762:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -2277.436057145762:\\
\;\;\;\;0\\

\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 (<= a -2277.436057145762) 0.0 (/ 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 (a <= -2277.436057145762) {
		tmp = 0.0;
	} 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
Herbie0.8
\[\frac{1}{1 + e^{b - a}}\]

Alternatives

Alternative 1
Error0.7
Cost19520
\[\frac{e^{a}}{e^{b} + e^{a}}\]
Alternative 2
Error12.3
Cost1090
\[\begin{array}{l} \mathbf{if}\;b \leq -8.261657521506199 \cdot 10^{-07}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.3930900108874062 \cdot 10^{-18}:\\ \;\;\;\;0.5 + 0.25 \cdot \left(a - b\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 3
Error12.3
Cost962
\[\begin{array}{l} \mathbf{if}\;b \leq -8.261657521506199 \cdot 10^{-07}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 6.941599052022013 \cdot 10^{-19}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 4
Error22.4
Cost385
\[\begin{array}{l} \mathbf{if}\;b \leq -370873287.4583278:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 5
Error46.5
Cost64
\[1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if a < -2277.4360571457619

    1. Initial program 0.2

      \[0\]

    if -2277.4360571457619 < a

    1. Initial program 0.8

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2277.436057145762:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))