Average Error: 0.0 → 0.5
Time: 1.0s
Precision: binary64
\[e^{-\left(1 - x \cdot x\right)} \]
↓
\[e^{-1} \cdot \mathsf{fma}\left(x, x, 1\right) \]
(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))
↓
(FPCore (x) :precision binary64 (* (exp -1.0) (fma x x 1.0)))
double code(double x) {
	return exp(-(1.0 - (x * x)));
}
↓
double code(double x) {
	return exp(-1.0) * fma(x, x, 1.0);
}

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

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

e^{-\left(1 - x \cdot x\right)}
↓
e^{-1} \cdot \mathsf{fma}\left(x, x, 1\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. Taylor expanded in x around 0 0.5

    \[\leadsto \color{blue}{e^{-1} + e^{-1} \cdot {x}^{2}} \]

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2022159 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))