exp neg sub

Average Error: 0.0 → 0.0

Time: 18.9s

Precision: 64

Math

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

↓

\[\left(\sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}} \cdot \left(\sqrt{e^{-1}} \cdot \sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}}\right)\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{e^{-1}}\right)\]

C

double f(double x) {
        double r41555 = 1.0;
        double r41556 = x;
        double r41557 = r41556 * r41556;
        double r41558 = r41555 - r41557;
        double r41559 = -r41558;
        double r41560 = exp(r41559);
        return r41560;
}

↓

double f(double x) {
        double r41561 = x;
        double r41562 = exp(r41561);
        double r41563 = 2.0;
        double r41564 = r41561 / r41563;
        double r41565 = pow(r41562, r41564);
        double r41566 = sqrt(r41565);
        double r41567 = 1.0;
        double r41568 = -r41567;
        double r41569 = exp(r41568);
        double r41570 = sqrt(r41569);
        double r41571 = r41570 * r41566;
        double r41572 = r41566 * r41571;
        double r41573 = r41565 * r41570;
        double r41574 = r41572 * r41573;
        return r41574;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[e^{-\left(1 - x \cdot x\right)}\]
2. Using strategy rm

3. Applied sub-neg 0.0

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

4. Applied distribute-neg-in 0.0

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

5. Applied exp-sum 0.0

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

6. Simplified 0.0

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

8. Applied sqr-pow 0.0

   \[\leadsto e^{-1} \cdot \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}\]

9. Applied add-sqr-sqrt 0.0

   \[\leadsto \color{blue}{\left(\sqrt{e^{-1}} \cdot \sqrt{e^{-1}}\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)\]

10. Applied unswap-sqr 0.0

    \[\leadsto \color{blue}{\left(\sqrt{e^{-1}} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \left(\sqrt{e^{-1}} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}\]

11. Simplified 0.0

    \[\leadsto \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{e^{-1}}\right)} \cdot \left(\sqrt{e^{-1}} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)\]

12. Simplified 0.0

    \[\leadsto \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{e^{-1}}\right) \cdot \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{e^{-1}}\right)}\]
13. Using strategy rm

14. Applied add-sqr-sqrt 0.0

    \[\leadsto \left(\color{blue}{\left(\sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}} \cdot \sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}}\right)} \cdot \sqrt{e^{-1}}\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{e^{-1}}\right)\]

15. Applied associate-*l* 0.0

    \[\leadsto \color{blue}{\left(\sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}} \cdot \left(\sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}} \cdot \sqrt{e^{-1}}\right)\right)} \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{e^{-1}}\right)\]

16. Final simplification 0.0

    \[\leadsto \left(\sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}} \cdot \left(\sqrt{e^{-1}} \cdot \sqrt{{\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}}\right)\right) \cdot \left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot \sqrt{e^{-1}}\right)\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1.0 (* x x)))))