Quotient of sum of exps

Average Error: 0.8 → 0.8

Time: 26.4s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = 2.4112655861716263e+231
  2. b = -5.670122129476186e-07

Math

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

↓

\[\frac{\sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}} \cdot \sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}}\]

C

double f(double a, double b) {
        double r187305 = a;
        double r187306 = exp(r187305);
        double r187307 = b;
        double r187308 = exp(r187307);
        double r187309 = r187306 + r187308;
        double r187310 = r187306 / r187309;
        return r187310;
}

↓

double f(double a, double b) {
        double r187311 = 1.0;
        double r187312 = a;
        double r187313 = exp(r187312);
        double r187314 = b;
        double r187315 = exp(r187314);
        double r187316 = r187313 + r187315;
        double r187317 = sqrt(r187316);
        double r187318 = cbrt(r187317);
        double r187319 = r187318 * r187318;
        double r187320 = r187311 / r187319;
        double r187321 = sqrt(r187320);
        double r187322 = r187313 / r187318;
        double r187323 = r187317 / r187322;
        double r187324 = cbrt(r187323);
        double r187325 = r187324 * r187324;
        double r187326 = r187321 / r187325;
        double r187327 = r187321 / r187324;
        double r187328 = r187326 * r187327;
        return r187328;
}

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-sqr-sqrt 1.3

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

4. Applied associate-/r* 1.1

   \[\leadsto \color{blue}{\frac{\frac{e^{a}}{\sqrt{e^{a} + e^{b}}}}{\sqrt{e^{a} + e^{b}}}}\]
5. Using strategy rm

6. Applied add-cube-cbrt 1.1

   \[\leadsto \frac{\frac{e^{a}}{\color{blue}{\left(\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}\right) \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt{e^{a} + e^{b}}}\]

7. Applied *-un-lft-identity 1.1

   \[\leadsto \frac{\frac{\color{blue}{1 \cdot e^{a}}}{\left(\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}\right) \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}{\sqrt{e^{a} + e^{b}}}\]

8. Applied times-frac 1.1

   \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}} \cdot \frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt{e^{a} + e^{b}}}\]

9. Applied associate-/l* 1.1

   \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}}\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.8

    \[\leadsto \frac{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}{\color{blue}{\left(\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}} \cdot \sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}\right) \cdot \sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}}}\]

12. Applied add-sqr-sqrt 0.8

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}}{\left(\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}} \cdot \sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}\right) \cdot \sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}}\]

13. Applied times-frac 0.8

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}} \cdot \sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}}}\]

14. Final simplification 0.8

    \[\leadsto \frac{\sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}} \cdot \sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt[3]{\sqrt{e^{a} + e^{b}}} \cdot \sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}{\sqrt[3]{\frac{\sqrt{e^{a} + e^{b}}}{\frac{e^{a}}{\sqrt[3]{\sqrt{e^{a} + e^{b}}}}}}}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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