exp neg sub

Average Error: 0.0 → 0.0

Time: 21.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r55397 = 1.0;
        double r55398 = x;
        double r55399 = r55398 * r55398;
        double r55400 = r55397 - r55399;
        double r55401 = -r55400;
        double r55402 = exp(r55401);
        return r55402;
}

↓

double f(double x) {
        double r55403 = 1.0;
        double r55404 = sqrt(r55403);
        double r55405 = -r55404;
        double r55406 = exp(r55405);
        double r55407 = x;
        double r55408 = r55404 - r55407;
        double r55409 = pow(r55406, r55408);
        double r55410 = sqrt(r55409);
        double r55411 = -r55407;
        double r55412 = exp(r55411);
        double r55413 = pow(r55412, r55408);
        double r55414 = sqrt(r55413);
        double r55415 = r55404 + r55407;
        double r55416 = -r55415;
        double r55417 = exp(r55416);
        double r55418 = pow(r55417, r55408);
        double r55419 = sqrt(r55418);
        double r55420 = r55414 * r55419;
        double r55421 = r55410 * r55420;
        return r55421;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-sqr-sqrt 0.0

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

4. Applied difference-of-squares 0.0

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

5. Applied distribute-lft-neg-in 0.0

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

6. Applied exp-prod 0.0

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

8. Applied add-sqr-sqrt 0.0

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

10. Applied distribute-neg-in 0.0

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

11. Applied exp-sum 0.0

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

12. Applied unpow-prod-down 0.0

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

13. Applied sqrt-prod 0.0

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

14. Applied associate-*l* 0.0

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

15. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1.0 (* x x)))))