Average Error: 0.0 → 0.0
Time: 17.6s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[{\left(\sqrt{e^{x}}\right)}^{x} \cdot \left({\left(\sqrt{e^{x}}\right)}^{x} \cdot e^{-1}\right)\]
e^{-\left(1 - x \cdot x\right)}
↓
{\left(\sqrt{e^{x}}\right)}^{x} \cdot \left({\left(\sqrt{e^{x}}\right)}^{x} \cdot e^{-1}\right)
double f(double x) {
        double r1122466 = 1.0;
        double r1122467 = x;
        double r1122468 = r1122467 * r1122467;
        double r1122469 = r1122466 - r1122468;
        double r1122470 = -r1122469;
        double r1122471 = exp(r1122470);
        return r1122471;
}
↓
double f(double x) {
        double r1122472 = x;
        double r1122473 = exp(r1122472);
        double r1122474 = sqrt(r1122473);
        double r1122475 = pow(r1122474, r1122472);
        double r1122476 = -1.0;
        double r1122477 = exp(r1122476);
        double r1122478 = r1122475 * r1122477;
        double r1122479 = r1122475 * r1122478;
        return r1122479;
}

Error

Bits error versus x

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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 fma-udef0.0

    \[\leadsto e^{\color{blue}{x \cdot x + -1}}\]

  5. Applied exp-sum0.0

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

  7. Applied exp-prod0.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Applied unpow-prod-down0.0

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

  11. Applied associate-*l*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))