Quotient of sum of exps

Average Error: 0.7 → 0.7

Time: 2.5s

Precision: 64

• could not determine a ground truth for program body

  1. a = -2.3323497117317315e+256
  2. b = -1.3537835572354477e+237

Math

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

↓

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

C

double code(double a, double b) {
	return (exp(a) / (exp(a) + exp(b)));
}

↓

double code(double a, double b) {
	return (exp(a) / (exp(a) + exp(b)));
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.7
Target	0.0
Herbie	0.7

\[\frac{1}{1 + e^{b - a}}\]

Derivation

1. Initial program 0.7

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

2. Final simplification 0.7

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1 (+ 1 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))