Average Error: 0.0 → 0.0
Time: 1.1s
Precision: binary64
\[e^{-\left(1 - x \cdot x\right)} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(x, x, -1\right)}\right)\right) \]
e^{-\left(1 - x \cdot x\right)}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(x, x, -1\right)}\right)\right)

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))
↓
(FPCore (x) :precision binary64 (expm1 (log1p (exp (fma x x -1.0)))))
double code(double x) {
	return exp(-(1.0 - (x * x)));
}
↓
double code(double x) {
	return expm1(log1p(exp(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 expm1-log1p-u_binary640.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

  4. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(x, x, -1\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022019 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))