Average Error: 0.0 → 0.0
Time: 13.4s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[e^{x \cdot x} \cdot \frac{1}{e}\]
e^{-\left(1 - x \cdot x\right)}
↓
e^{x \cdot x} \cdot \frac{1}{e}
double f(double x) {
        double r880440 = 1.0;
        double r880441 = x;
        double r880442 = r880441 * r880441;
        double r880443 = r880440 - r880442;
        double r880444 = -r880443;
        double r880445 = exp(r880444);
        return r880445;
}
↓
double f(double x) {
        double r880446 = x;
        double r880447 = r880446 * r880446;
        double r880448 = exp(r880447);
        double r880449 = 1.0;
        double r880450 = exp(1.0);
        double r880451 = r880449 / r880450;
        double r880452 = r880448 * r880451;
        return r880452;
}

Error

Bits error versus x

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Results

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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. Simplified0.0

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

  7. Final simplification0.0

    \[\leadsto e^{x \cdot x} \cdot \frac{1}{e}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))