Average Error: 0.6 → 0.3
Time: 2.7s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 1.769608985345696 \cdot 10^{+308}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{e^{a}}\\ \frac{e^{a}}{\mathsf{fma}\left(t_0, t_0, e^{b}\right)} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\begin{array}{l}
\mathbf{if}\;e^{a} \leq 1.769608985345696 \cdot 10^{+308}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt{e^{a}}\\
\frac{e^{a}}{\mathsf{fma}\left(t_0, t_0, e^{b}\right)}
\end{array}\\

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


\end{array}

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1.769608985345696e+308)
   (let* ((t_0 (sqrt (exp a)))) (/ (exp a) (fma t_0 t_0 (exp b))))
   (/ 1.0 (+ (exp b) 1.0))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1.769608985345696e+308) {
		double t_0_1 = sqrt(exp(a));
		tmp = exp(a) / fma(t_0_1, t_0_1, exp(b));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.6
Target0.0
Herbie0.3
\[\frac{1}{1 + e^{b - a}} \]

Derivation

  1. Split input into 2 regimes

  2. if (exp.f64 a) < 1.76960898534569605e308

    1. Initial program 0.2

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

    2. Applied add-sqr-sqrt_binary640.2

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

    3. Applied fma-def_binary640.2

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}} \]

    if 1.76960898534569605e308 < (exp.f64 a)

    1. Initial program 64.0

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

    2. Taylor expanded in a around 0 23.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 1.769608985345696 \cdot 10^{+308}:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\sqrt{e^{a}}, \sqrt{e^{a}}, e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2021357 
(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))))