Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[\frac{{\left(\sqrt{\sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}} \cdot \frac{{\left(\sqrt{e^{x}}\right)}^{x}}{\frac{\sqrt{e^{1}}}{{\left(\left|\sqrt[3]{e^{x}}\right|\right)}^{x}}}\]
e^{-\left(1 - x \cdot x\right)}
↓
\frac{{\left(\sqrt{\sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}} \cdot \frac{{\left(\sqrt{e^{x}}\right)}^{x}}{\frac{\sqrt{e^{1}}}{{\left(\left|\sqrt[3]{e^{x}}\right|\right)}^{x}}}
double f(double x) {
        double r49793 = 1.0;
        double r49794 = x;
        double r49795 = r49794 * r49794;
        double r49796 = r49793 - r49795;
        double r49797 = -r49796;
        double r49798 = exp(r49797);
        return r49798;
}
↓
double f(double x) {
        double r49799 = x;
        double r49800 = exp(r49799);
        double r49801 = cbrt(r49800);
        double r49802 = sqrt(r49801);
        double r49803 = pow(r49802, r49799);
        double r49804 = 1.0;
        double r49805 = exp(r49804);
        double r49806 = sqrt(r49805);
        double r49807 = r49803 / r49806;
        double r49808 = sqrt(r49800);
        double r49809 = pow(r49808, r49799);
        double r49810 = fabs(r49801);
        double r49811 = pow(r49810, r49799);
        double r49812 = r49806 / r49811;
        double r49813 = r49809 / r49812;
        double r49814 = r49807 * r49813;
        return r49814;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

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

    \[\leadsto e^{\color{blue}{x \cdot x + \left(-1\right)}}\]

  5. Applied exp-sum0.0

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

  6. Simplified0.0

    \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot e^{-1}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.0

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

  9. Applied unpow-prod-down0.0

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

  10. Applied associate-*l*0.0

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

  11. Simplified0.0

    \[\leadsto {\left(\sqrt{e^{x}}\right)}^{x} \cdot \color{blue}{\frac{{\left(\sqrt{e^{x}}\right)}^{x}}{e^{1}}}\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt1.0

    \[\leadsto {\left(\sqrt{e^{x}}\right)}^{x} \cdot \frac{{\left(\sqrt{e^{x}}\right)}^{x}}{\color{blue}{\sqrt{e^{1}} \cdot \sqrt{e^{1}}}}\]

  14. Applied add-cube-cbrt1.0

    \[\leadsto {\left(\sqrt{e^{x}}\right)}^{x} \cdot \frac{{\left(\sqrt{\color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}}}\right)}^{x}}{\sqrt{e^{1}} \cdot \sqrt{e^{1}}}\]

  15. Applied sqrt-prod1.0

    \[\leadsto {\left(\sqrt{e^{x}}\right)}^{x} \cdot \frac{{\color{blue}{\left(\sqrt{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt{\sqrt[3]{e^{x}}}\right)}}^{x}}{\sqrt{e^{1}} \cdot \sqrt{e^{1}}}\]

  16. Applied unpow-prod-down1.0

    \[\leadsto {\left(\sqrt{e^{x}}\right)}^{x} \cdot \frac{\color{blue}{{\left(\sqrt{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}\right)}^{x} \cdot {\left(\sqrt{\sqrt[3]{e^{x}}}\right)}^{x}}}{\sqrt{e^{1}} \cdot \sqrt{e^{1}}}\]

  17. Applied times-frac0.0

    \[\leadsto {\left(\sqrt{e^{x}}\right)}^{x} \cdot \color{blue}{\left(\frac{{\left(\sqrt{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}} \cdot \frac{{\left(\sqrt{\sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}}\right)}\]

  18. Applied associate-*r*0.0

    \[\leadsto \color{blue}{\left({\left(\sqrt{e^{x}}\right)}^{x} \cdot \frac{{\left(\sqrt{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}}\right) \cdot \frac{{\left(\sqrt{\sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}}}\]

  19. Simplified0.0

    \[\leadsto \color{blue}{\frac{{\left(\sqrt{e^{x}}\right)}^{x}}{\frac{\sqrt{e^{1}}}{{\left(\left|\sqrt[3]{e^{x}}\right|\right)}^{x}}}} \cdot \frac{{\left(\sqrt{\sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}}\]

  20. Final simplification0.0

    \[\leadsto \frac{{\left(\sqrt{\sqrt[3]{e^{x}}}\right)}^{x}}{\sqrt{e^{1}}} \cdot \frac{{\left(\sqrt{e^{x}}\right)}^{x}}{\frac{\sqrt{e^{1}}}{{\left(\left|\sqrt[3]{e^{x}}\right|\right)}^{x}}}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1.0 (* x x)))))