Average Error: 0.0 → 0.0
Time: 3.7s
Precision: binary64
\[e^{-\left(1 - x \cdot x\right)} \]
↓
\[e^{\log e \cdot \mathsf{fma}\left(x, x, -1\right)} \]
e^{-\left(1 - x \cdot x\right)}
↓
e^{\log e \cdot \mathsf{fma}\left(x, x, -1\right)}

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

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. 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 pow-to-exp_binary640.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022055 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))