Average Error: 0.0 → 0.0
Time: 19.4s
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 r34376 = 1.0;
        double r34377 = x;
        double r34378 = r34377 * r34377;
        double r34379 = r34376 - r34378;
        double r34380 = -r34379;
        double r34381 = exp(r34380);
        return r34381;
}
↓
double f(double x) {
        double r34382 = x;
        double r34383 = exp(r34382);
        double r34384 = pow(r34383, r34382);
        double r34385 = 1.0;
        double r34386 = -r34385;
        double r34387 = exp(r34386);
        double r34388 = r34384 * r34387;
        return r34388;
}

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 *-un-lft-identity0.0

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

  5. Applied exp-prod0.0

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

  6. Simplified0.0

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

  8. Applied add-sqr-sqrt0.0

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  12. Applied fma-udef0.0

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

  13. Applied exp-sum0.0

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

  14. Applied sqrt-prod0.0

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

  15. Applied fma-udef0.0

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

  16. Applied exp-sum0.0

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

  17. Applied sqrt-prod0.0

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

  18. Applied swap-sqr0.0

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

  19. Simplified0.0

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

  20. Simplified0.0

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

  21. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1 (* x x)))))