exp neg sub

Average Error: 0.0 → 0.1

Time: 8.5s

Precision: 64

Math

\[e^{-\left(1 - x \cdot x\right)}\]

↓

\[{\left(e^{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}\]

C

double f(double x) {
        double r1980601 = 1.0;
        double r1980602 = x;
        double r1980603 = r1980602 * r1980602;
        double r1980604 = r1980601 - r1980603;
        double r1980605 = -r1980604;
        double r1980606 = exp(r1980605);
        return r1980606;
}

↓

double f(double x) {
        double r1980607 = x;
        double r1980608 = -1.0;
        double r1980609 = fma(r1980607, r1980607, r1980608);
        double r1980610 = cbrt(r1980609);
        double r1980611 = r1980610 * r1980610;
        double r1980612 = exp(r1980611);
        double r1980613 = pow(r1980612, r1980610);
        return r1980613;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[e^{-\left(1 - x \cdot x\right)}\]

2. Simplified 0.0

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

4. Applied add-cube-cbrt 0.1

   \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}}\]

5. Applied exp-prod 0.1

   \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}}\]

6. Final simplification 0.1

   \[\leadsto {\left(e^{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))