exp neg sub

Average Accuracy: 100.0% → 100.0%

Time: 3.7s

Precision: binary64

Cost: 13120

Math

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

↓

\[1 + \mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (+ 1.0 (expm1 (fma x x -1.0))))

C

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

↓

double code(double x) {
	return 1.0 + expm1(fma(x, x, -1.0));
}

Julia

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

↓

function code(x)
	return Float64(1.0 + expm1(fma(x, x, -1.0)))
end

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ e^{-\left(1 - x \cdot x\right)} \]
   neg-sub0 [=>] 100.0	\[ e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]
   associate--r- [=>] 100.0	\[ e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]
   metadata-eval [=>] 100.0	\[ e^{\color{blue}{-1} + x \cdot x} \]
   +-commutative [=>] 100.0	\[ e^{\color{blue}{x \cdot x + -1}} \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ e^{x \cdot x + -1} \]
   *-un-lft-identity [=>] 100.0	\[ e^{\color{blue}{1 \cdot \left(x \cdot x + -1\right)}} \]
   exp-prod [=>] 100.0	\[ \color{blue}{{\left(e^{1}\right)}^{\left(x \cdot x + -1\right)}} \]
   fma-def [=>] 100.0	\[ {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}} \]

4. Simplified 100.0%

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

   [Start] 100.0	\[ {\left(e^{1}\right)}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]
   exp-1-e [=>] 100.0	\[ {\color{blue}{e}}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{1 + \mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]
   Proof

   [Start] 100.0	\[ {e}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]
   e-exp-1 [=>] 100.0	\[ {\color{blue}{\left(e^{1}\right)}}^{\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]
   exp-prod [<=] 100.0	\[ \color{blue}{e^{1 \cdot \mathsf{fma}\left(x, x, -1\right)}} \]
   *-un-lft-identity [<=] 100.0	\[ e^{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]
   log1p-expm1-u [=>] 100.0	\[ e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right)\right)}} \]
   log1p-udef [=>] 100.0	\[ e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right)\right)}} \]
   add-exp-log [<=] 100.0	\[ \color{blue}{1 + \mathsf{expm1}\left(\mathsf{fma}\left(x, x, -1\right)\right)} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6720

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

Alternative 2
Accuracy	98.5%
Cost	6464

\[e^{-1} \]

Reproduce

herbie shell --seed 2023126 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))