Quotient of sum of exps

Average Error: 0.6 → 0.5

Time: 3.8s

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 r168015 = a;
        double r168016 = exp(r168015);
        double r168017 = b;
        double r168018 = exp(r168017);
        double r168019 = r168016 + r168018;
        double r168020 = r168016 / r168019;
        return r168020;
}

↓

double f(double a, double b) {
        double r168021 = a;
        double r168022 = exp(r168021);
        double r168023 = b;
        double r168024 = exp(r168023);
        double r168025 = r168022 + r168024;
        double r168026 = log(r168025);
        double r168027 = r168021 - r168026;
        double r168028 = exp(r168027);
        double r168029 = log1p(r168028);
        double r168030 = expm1(r168029);
        return r168030;
}

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))))