exp neg sub

Average Error: 0.0 → 0.0

Time: 2.6s

Precision: binary64

Cost: 19392

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

↓

(FPCore (x) :precision binary64 (pow E (fma x x -1.0)))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

↓

double code(double x) {
	return pow(((double) M_E), fma(x, x, -1.0));
}

Julia

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

↓

function code(x)
	return exp(1) ^ fma(x, x, -1.0)
end

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ e^{-\left(1 - x \cdot x\right)} \]
   neg-sub0 [=>] 0.0	\[ e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]
   associate--r- [=>] 0.0	\[ e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]
   metadata-eval [=>] 0.0	\[ e^{\color{blue}{-1} + x \cdot x} \]
   +-commutative [=>] 0.0	\[ e^{\color{blue}{x \cdot x + -1}} \]

3. Applied egg-rr 0.0

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

4. Simplified 0.0

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

   [Start] 0.0	\[ {\left(e^{1}\right)}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]
   exp-1-e [=>] 0.0	\[ {\color{blue}{e}}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6720

\[e^{-1 + x \cdot x} \]

Alternative 2
Error	0.9
Cost	6464

\[e^{-1} \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))