exp neg sub

Average Accuracy: 50.4% → 51.1%

Time: 7.2s

Precision: binary64

Cost: 32896

Math

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

↓

\[{\left({\left(e^{1 + {x}^{3}}\right)}^{\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right) - x}\right)}\right)}^{\left(x + -1\right)} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (pow (pow (exp (+ 1.0 (pow x 3.0))) (/ 1.0 (- (fma x x 1.0) x))) (+ x -1.0)))

C

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

↓

double code(double x) {
	return pow(pow(exp((1.0 + pow(x, 3.0))), (1.0 / (fma(x, x, 1.0) - x))), (x + -1.0));
}

Julia

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

↓

function code(x)
	return (exp(Float64(1.0 + (x ^ 3.0))) ^ Float64(1.0 / Float64(fma(x, x, 1.0) - x))) ^ Float64(x + -1.0)
end

Wolfram

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

↓

code[x_] := N[Power[N[Power[N[Exp[N[(1.0 + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.4%

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

2. Simplified 50.4%

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

   [Start] 50.4	\[ e^{-\left(1 - x \cdot x\right)} \]
   neg-sub0 [=>] 50.4	\[ e^{\color{blue}{0 - \left(1 - x \cdot x\right)}} \]
   associate--r- [=>] 50.4	\[ e^{\color{blue}{\left(0 - 1\right) + x \cdot x}} \]
   metadata-eval [=>] 50.4	\[ e^{\color{blue}{-1} + x \cdot x} \]
   +-commutative [=>] 50.4	\[ e^{\color{blue}{x \cdot x + -1}} \]

3. Applied egg-rr 50.4%

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

   [Start] 50.4	\[ e^{x \cdot x + -1} \]
   difference-of-sqr--1 [=>] 50.4	\[ e^{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]
   exp-prod [=>] 50.4	\[ \color{blue}{{\left(e^{x + 1}\right)}^{\left(x - 1\right)}} \]
   sub-neg [=>] 50.4	\[ {\left(e^{x + 1}\right)}^{\color{blue}{\left(x + \left(-1\right)\right)}} \]
   metadata-eval [=>] 50.4	\[ {\left(e^{x + 1}\right)}^{\left(x + \color{blue}{-1}\right)} \]

4. Applied egg-rr 51.1%

   \[\leadsto {\color{blue}{\left({\left(e^{1 + {x}^{3}}\right)}^{\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right) - x}\right)}\right)}}^{\left(x + -1\right)} \]
   Proof

   [Start] 50.4	\[ {\left(e^{x + 1}\right)}^{\left(x + -1\right)} \]
   flip3-+ [=>] 50.4	\[ {\left(e^{\color{blue}{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}}\right)}^{\left(x + -1\right)} \]
   div-inv [=>] 50.4	\[ {\left(e^{\color{blue}{\left({x}^{3} + {1}^{3}\right) \cdot \frac{1}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}}\right)}^{\left(x + -1\right)} \]
   add-log-exp [=>] 50.3	\[ {\left(e^{\color{blue}{\log \left(e^{{x}^{3} + {1}^{3}}\right)} \cdot \frac{1}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}\right)}^{\left(x + -1\right)} \]
   exp-to-pow [=>] 51.1	\[ {\color{blue}{\left({\left(e^{{x}^{3} + {1}^{3}}\right)}^{\left(\frac{1}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}\right)}}^{\left(x + -1\right)} \]
   metadata-eval [=>] 51.1	\[ {\left({\left(e^{{x}^{3} + \color{blue}{1}}\right)}^{\left(\frac{1}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}\right)}^{\left(x + -1\right)} \]
   +-commutative [=>] 51.1	\[ {\left({\left(e^{\color{blue}{1 + {x}^{3}}}\right)}^{\left(\frac{1}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}\right)}^{\left(x + -1\right)} \]
   metadata-eval [=>] 51.1	\[ {\left({\left(e^{1 + {x}^{3}}\right)}^{\left(\frac{1}{x \cdot x + \left(\color{blue}{1} - x \cdot 1\right)}\right)}\right)}^{\left(x + -1\right)} \]
   *-rgt-identity [=>] 51.1	\[ {\left({\left(e^{1 + {x}^{3}}\right)}^{\left(\frac{1}{x \cdot x + \left(1 - \color{blue}{x}\right)}\right)}\right)}^{\left(x + -1\right)} \]
   associate-+r- [=>] 51.1	\[ {\left({\left(e^{1 + {x}^{3}}\right)}^{\left(\frac{1}{\color{blue}{\left(x \cdot x + 1\right) - x}}\right)}\right)}^{\left(x + -1\right)} \]
   fma-def [=>] 51.1	\[ {\left({\left(e^{1 + {x}^{3}}\right)}^{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)} - x}\right)}\right)}^{\left(x + -1\right)} \]

5. Final simplification 51.1%

   \[\leadsto {\left({\left(e^{1 + {x}^{3}}\right)}^{\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right) - x}\right)}\right)}^{\left(x + -1\right)} \]

Alternatives

Alternative 1
Accuracy	51.3%
Cost	6848

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

Alternative 2
Accuracy	50.4%
Cost	6720

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

Alternative 3
Accuracy	51.3%
Cost	6464

\[e^{-1} \]

Alternative 4
Accuracy	3.5%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023153 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))