Quotient of sum of exps

Average Error: 0.8 → 0.8

Time: 9.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. a = -1.3538666308402146e+249
  2. b = -1.3236084789187749e+62

Math

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

↓

\[\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)\]

C

double f(double a, double b) {
        double r114140 = a;
        double r114141 = exp(r114140);
        double r114142 = b;
        double r114143 = exp(r114142);
        double r114144 = r114141 + r114143;
        double r114145 = r114141 / r114144;
        return r114145;
}

↓

double f(double a, double b) {
        double r114146 = a;
        double r114147 = exp(r114146);
        double r114148 = exp(r114147);
        double r114149 = 1.0;
        double r114150 = b;
        double r114151 = exp(r114150);
        double r114152 = r114147 + r114151;
        double r114153 = r114149 / r114152;
        double r114154 = pow(r114148, r114153);
        double r114155 = log(r114154);
        return r114155;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.8
Target	0.0
Herbie	0.8

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

Derivation

1. Initial program 0.8

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

3. Applied add-log-exp 0.9

   \[\leadsto \color{blue}{\log \left(e^{\frac{e^{a}}{e^{a} + e^{b}}}\right)}\]
4. Using strategy rm

5. Applied div-inv 0.9

   \[\leadsto \log \left(e^{\color{blue}{e^{a} \cdot \frac{1}{e^{a} + e^{b}}}}\right)\]

6. Applied exp-prod 0.8

   \[\leadsto \log \color{blue}{\left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)}\]

7. Final simplification 0.8

   \[\leadsto \log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right)\]

Reproduce

herbie shell --seed 2019350 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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