Average Error: 0.0 → 0.0
Time: 17.6s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[e^{\mathsf{fma}\left(x, x, -1\right)}\]
e^{-\left(1 - x \cdot x\right)}
↓
e^{\mathsf{fma}\left(x, x, -1\right)}
double f(double x) {
        double r1116413 = 1.0;
        double r1116414 = x;
        double r1116415 = r1116414 * r1116414;
        double r1116416 = r1116413 - r1116415;
        double r1116417 = -r1116416;
        double r1116418 = exp(r1116417);
        return r1116418;
}
↓
double f(double x) {
        double r1116419 = x;
        double r1116420 = -1.0;
        double r1116421 = fma(r1116419, r1116419, r1116420);
        double r1116422 = exp(r1116421);
        return r1116422;
}

Error

Bits error versus x

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{e^{{x}^{2}} \cdot e^{-1}}\]

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))