Average Error: 0.0 → 0.0
Time: 35.3s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[\sqrt[3]{{e}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)}} \cdot \left(\left(\left(\sqrt{\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}}\right) \cdot \left(\sqrt{\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}}\right)\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(x, x, -1\right)}}\right)\]
e^{-\left(1 - x \cdot x\right)}
↓
\sqrt[3]{{e}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)}} \cdot \left(\left(\left(\sqrt{\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}}\right) \cdot \left(\sqrt{\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{\mathsf{fma}\left(x, x, -1\right)}}}}\right)\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(x, x, -1\right)}}\right)
double f(double x) {
        double r4167164 = 1.0;
        double r4167165 = x;
        double r4167166 = r4167165 * r4167165;
        double r4167167 = r4167164 - r4167166;
        double r4167168 = -r4167167;
        double r4167169 = exp(r4167168);
        return r4167169;
}
↓
double f(double x) {
        double r4167170 = exp(1.0);
        double r4167171 = x;
        double r4167172 = -1.0;
        double r4167173 = fma(r4167171, r4167171, r4167172);
        double r4167174 = pow(r4167170, r4167173);
        double r4167175 = cbrt(r4167174);
        double r4167176 = exp(r4167173);
        double r4167177 = sqrt(r4167176);
        double r4167178 = cbrt(r4167177);
        double r4167179 = sqrt(r4167178);
        double r4167180 = sqrt(r4167177);
        double r4167181 = cbrt(r4167180);
        double r4167182 = r4167179 * r4167181;
        double r4167183 = r4167182 * r4167182;
        double r4167184 = cbrt(r4167176);
        double r4167185 = r4167183 * r4167184;
        double r4167186 = r4167175 * r4167185;
        return r4167186;
}

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 add-cube-cbrt2.0

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

  6. Applied *-un-lft-identity2.0

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

  7. Applied exp-prod2.0

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

  8. Simplified2.0

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

  10. Applied add-sqr-sqrt2.0

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

  11. Applied cbrt-prod1.0

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

  13. Applied add-sqr-sqrt1.0

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

  14. Applied add-sqr-sqrt1.0

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

  15. Applied cbrt-prod0.0

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

  16. Applied unswap-sqr0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))