Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[\frac{\sqrt{{\left(e^{x}\right)}^{x}}}{\frac{e^{1}}{\sqrt{{\left(e^{x}\right)}^{x}}}}\]
e^{-\left(1 - x \cdot x\right)}
↓
\frac{\sqrt{{\left(e^{x}\right)}^{x}}}{\frac{e^{1}}{\sqrt{{\left(e^{x}\right)}^{x}}}}
double f(double x) {
        double r30432 = 1.0;
        double r30433 = x;
        double r30434 = r30433 * r30433;
        double r30435 = r30432 - r30434;
        double r30436 = -r30435;
        double r30437 = exp(r30436);
        return r30437;
}
↓
double f(double x) {
        double r30438 = x;
        double r30439 = exp(r30438);
        double r30440 = pow(r30439, r30438);
        double r30441 = sqrt(r30440);
        double r30442 = 1.0;
        double r30443 = exp(r30442);
        double r30444 = r30443 / r30441;
        double r30445 = r30441 / r30444;
        return r30445;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-log-exp0.0

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

  4. Applied add-log-exp0.0

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

  5. Applied diff-log0.0

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

  6. Applied neg-log0.0

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

  7. Applied rem-exp-log0.0

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

  9. Applied add-log-exp0.0

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

  10. Applied exp-to-pow0.0

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

  12. Applied add-sqr-sqrt0.0

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

  13. Applied *-un-lft-identity0.0

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

  14. Applied times-frac0.0

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

  15. Applied associate-/r*0.0

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

  16. Simplified0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1 (* x x)))))