Average Error: 0.0 → 0.0
Time: 7.3s
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 r654258 = 1.0;
        double r654259 = x;
        double r654260 = r654259 * r654259;
        double r654261 = r654258 - r654260;
        double r654262 = -r654261;
        double r654263 = exp(r654262);
        return r654263;
}
↓
double f(double x) {
        double r654264 = x;
        double r654265 = r654264 * r654264;
        double r654266 = exp(r654265);
        double r654267 = exp(1.0);
        double r654268 = r654266 / r654267;
        return r654268;
}

Error

Bits error versus x

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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. Simplified0.0

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

  8. Applied associate-*r/0.0

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

  9. Simplified0.0

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

  10. 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)))))