exp neg sub

Average Error: 0.0 → 0.0

Time: 3.4s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[e^{-\left(1 - x \cdot x\right)}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.0

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

4. Applied difference-of-squares 0.0

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

5. Applied distribute-lft-neg-in 0.0

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

6. Applied exp-prod 0.0

   \[\leadsto \color{blue}{{\left(e^{-\left(\sqrt{1} + x\right)}\right)}^{\left(\sqrt{1} - x\right)}}\]
7. Using strategy rm

8. Applied distribute-neg-in 0.0

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

9. Applied exp-sum 0.0

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

10. Applied unpow-prod-down 0.0

    \[\leadsto \color{blue}{{\left(e^{-\sqrt{1}}\right)}^{\left(\sqrt{1} - x\right)} \cdot {\left(e^{-x}\right)}^{\left(\sqrt{1} - x\right)}}\]
11. Using strategy rm

12. Applied pow-exp 0.0

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

13. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1 (* x x)))))