Average Error: 0.0 → 0.0
Time: 2.6s
Precision: binary64
\[e^{-\left(1 - x \cdot x\right)} \]
↓
\[{\left(\sqrt{e}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)} \cdot e^{-1} \]
e^{-\left(1 - x \cdot x\right)}
↓
{\left(\sqrt{e}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)} \cdot e^{-1}

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))
↓
(FPCore (x) :precision binary64 (* (pow (sqrt E) (* 2.0 (* x x))) (exp -1.0)))
double code(double x) {
	return exp(-(1.0 - (x * x)));
}
↓
double code(double x) {
	return pow(sqrt((double) M_E), (2.0 * (x * x))) * exp(-1.0);
}

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. Applied *-un-lft-identity_binary640.0

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

  4. Applied exp-prod_binary640.0

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

  5. Simplified0.0

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

  6. Applied add-sqr-sqrt_binary641.0

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

  7. Applied unpow-prod-down_binary640.0

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

  8. Applied fma-udef_binary640.0

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

  9. Applied unpow-prod-up_binary640.0

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

  10. Applied fma-udef_binary640.0

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

  11. Applied unpow-prod-up_binary640.0

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

  12. Applied swap-sqr_binary640.0

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

  13. Simplified0.0

    \[\leadsto \color{blue}{{\left(\sqrt{e}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)}} \cdot \left({\left(\sqrt{e}\right)}^{-1} \cdot {\left(\sqrt{e}\right)}^{-1}\right) \]

  14. Simplified0.0

    \[\leadsto {\left(\sqrt{e}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)} \cdot \color{blue}{e^{-1}} \]

  15. Final simplification0.0

    \[\leadsto {\left(\sqrt{e}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)} \cdot e^{-1} \]

Reproduce

herbie shell --seed 2022020 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))