Quotient of sum of exps

Average Error: 0.6 → 0.5

Time: 4.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = 4.595623646030552e+209
  2. b = -1.9489804220966382e-249

Math

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

↓

\[\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \log \left(e^{a} + e^{b}\right)}\right)\right)\]

C

double f(double a, double b) {
        double r158034 = a;
        double r158035 = exp(r158034);
        double r158036 = b;
        double r158037 = exp(r158036);
        double r158038 = r158035 + r158037;
        double r158039 = r158035 / r158038;
        return r158039;
}

↓

double f(double a, double b) {
        double r158040 = a;
        double r158041 = exp(r158040);
        double r158042 = b;
        double r158043 = exp(r158042);
        double r158044 = r158041 + r158043;
        double r158045 = log(r158044);
        double r158046 = r158040 - r158045;
        double r158047 = exp(r158046);
        double r158048 = log1p(r158047);
        double r158049 = expm1(r158048);
        return r158049;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.6
Target	0.0
Herbie	0.5

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

Derivation

1. Initial program 0.6

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

3. Applied add-exp-log 0.6

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

4. Applied div-exp 0.5

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

6. Applied expm1-log1p-u 0.5

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \log \left(e^{a} + e^{b}\right)}\right)\right)}\]

7. Final simplification 0.5

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \log \left(e^{a} + e^{b}\right)}\right)\right)\]

Reproduce

herbie shell --seed 2020056 +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))))