Average Error: 0.0 → 0.0
Time: 21.3s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[{\left(e^{x}\right)}^{x} \cdot e^{-1}\]
e^{-\left(1 - x \cdot x\right)}
↓
{\left(e^{x}\right)}^{x} \cdot e^{-1}
double f(double x) {
        double r39325 = 1.0;
        double r39326 = x;
        double r39327 = r39326 * r39326;
        double r39328 = r39325 - r39327;
        double r39329 = -r39328;
        double r39330 = exp(r39329);
        return r39330;
}
↓
double f(double x) {
        double r39331 = x;
        double r39332 = exp(r39331);
        double r39333 = pow(r39332, r39331);
        double r39334 = 1.0;
        double r39335 = -r39334;
        double r39336 = exp(r39335);
        double r39337 = r39333 * r39336;
        return r39337;
}

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. Simplified0.0

    \[\leadsto \color{blue}{e^{\mathsf{fma}\left(x, x, -1\right)}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

    \[\leadsto \color{blue}{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}\]
  5. Using strategy rm

  6. Applied fma-udef0.0

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

  7. Applied exp-sum0.0

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

  8. Applied sqrt-prod0.0

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

  9. Applied fma-udef0.0

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

  10. Applied exp-sum0.0

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

  11. Applied sqrt-prod0.0

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

  12. Applied swap-sqr0.0

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

  13. Simplified0.0

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

  14. Simplified0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1 (* x x)))))