Average Error: 0.0 → 0.0
Time: 2.9s
Precision: binary64
\[e^{-\left(1 - x \cdot x\right)} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(x, x, -1\right)}\right)\right) \]
(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))));
}

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end
↓
function code(x)
	return expm1(log1p(exp(fma(x, x, -1.0))))
end

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]
↓
code[x_] := N[(Exp[N[Log[1 + N[Exp[N[(x * x + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

e^{-\left(1 - x \cdot x\right)}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(x, x, -1\right)}\right)\right)

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 egg-rr0.0

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

  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 2022155 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))