Average Error: 0.0 → 0.0
Time: 6.6s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[\frac{e^{x \cdot x}}{e}\]
e^{-\left(1 - x \cdot x\right)}
↓
\frac{e^{x \cdot x}}{e}
double f(double x) {
        double r635765 = 1.0;
        double r635766 = x;
        double r635767 = r635766 * r635766;
        double r635768 = r635765 - r635767;
        double r635769 = -r635768;
        double r635770 = exp(r635769);
        return r635770;
}
↓
double f(double x) {
        double r635771 = x;
        double r635772 = r635771 * r635771;
        double r635773 = exp(r635772);
        double r635774 = exp(1.0);
        double r635775 = r635773 / r635774;
        return r635775;
}

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. Using strategy rm

  8. Applied un-div-inv0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))