Average Error: 0.6 → 0.6
Time: 3.1s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{e^{a}}\\ \frac{e^{a}}{\mathsf{fma}\left(t_0, t_0, e^{b}\right)} \end{array} \]
\frac{e^{a}}{e^{a} + e^{b}}
↓
\begin{array}{l}
t_0 := \sqrt{e^{a}}\\
\frac{e^{a}}{\mathsf{fma}\left(t_0, t_0, e^{b}\right)}
\end{array}

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (sqrt (exp a)))) (/ (exp a) (fma t_0 t_0 (exp b)))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	double t_0 = sqrt(exp(a));
	return exp(a) / fma(t_0, t_0, exp(b));
}

Error

Bits error versus a

Bits error versus b

Target

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

Derivation

  1. Initial program 0.6

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

  2. Applied add-sqr-sqrt_binary640.6

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

  3. Applied fma-def_binary640.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2022094 
(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))))