exp neg sub

Average Error: 0.0 → 0.0

Time: 18.8s

Precision: 64

Math

\[e^{-\left(1 - x \cdot x\right)}\]

↓

\[\left(\sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}}\right) \cdot \sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}}\]

C

double f(double x) {
        double r43423 = 1.0;
        double r43424 = x;
        double r43425 = r43424 * r43424;
        double r43426 = r43423 - r43425;
        double r43427 = -r43426;
        double r43428 = exp(r43427);
        return r43428;
}

↓

double f(double x) {
        double r43429 = x;
        double r43430 = 1.0;
        double r43431 = sqrt(r43430);
        double r43432 = r43429 + r43431;
        double r43433 = exp(r43432);
        double r43434 = sqrt(r43433);
        double r43435 = r43429 - r43431;
        double r43436 = pow(r43434, r43435);
        double r43437 = sqrt(r43436);
        double r43438 = pow(r43433, r43435);
        double r43439 = sqrt(r43438);
        double r43440 = r43437 * r43439;
        double r43441 = r43440 * r43437;
        return r43441;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[e^{-\left(1 - x \cdot x\right)}\]

2. Simplified 0.0

   \[\leadsto \color{blue}{e^{x \cdot x - 1}}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.0

   \[\leadsto e^{x \cdot x - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}\]

5. Applied difference-of-squares 0.0

   \[\leadsto e^{\color{blue}{\left(x + \sqrt{1}\right) \cdot \left(x - \sqrt{1}\right)}}\]

6. Applied exp-prod 0.0

   \[\leadsto \color{blue}{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.0

   \[\leadsto \color{blue}{\sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}}}\]
9. Using strategy rm

10. Applied add-sqr-sqrt 0.0

    \[\leadsto \sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{{\color{blue}{\left(\sqrt{e^{x + \sqrt{1}}} \cdot \sqrt{e^{x + \sqrt{1}}}\right)}}^{\left(x - \sqrt{1}\right)}}\]

11. Applied unpow-prod-down 0.0

    \[\leadsto \sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{\color{blue}{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)} \cdot {\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}}}\]

12. Applied sqrt-prod 0.0

    \[\leadsto \sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \color{blue}{\left(\sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}}\right)}\]

13. Applied associate-*r* 0.0

    \[\leadsto \color{blue}{\left(\sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}}\right) \cdot \sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}}}\]

14. Simplified 0.0

    \[\leadsto \color{blue}{\left(\sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}}\right)} \cdot \sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}}\]

15. Final simplification 0.0

    \[\leadsto \left(\sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}} \cdot \sqrt{{\left(e^{x + \sqrt{1}}\right)}^{\left(x - \sqrt{1}\right)}}\right) \cdot \sqrt{{\left(\sqrt{e^{x + \sqrt{1}}}\right)}^{\left(x - \sqrt{1}\right)}}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1.0 (* x x)))))