Average Error: 0.0 → 0.0
Time: 12.7s
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 r1518319 = 1.0;
        double r1518320 = x;
        double r1518321 = r1518320 * r1518320;
        double r1518322 = r1518319 - r1518321;
        double r1518323 = -r1518322;
        double r1518324 = exp(r1518323);
        return r1518324;
}
↓
double f(double x) {
        double r1518325 = x;
        double r1518326 = r1518325 * r1518325;
        double r1518327 = exp(r1518326);
        double r1518328 = 1.0;
        double r1518329 = exp(1.0);
        double r1518330 = r1518328 / r1518329;
        double r1518331 = r1518327 * r1518330;
        return r1518331;
}

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)))))