expfmod

Average Error: 59.8 → 59.8

Time: 8.5s

Precision: binary64

Math

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

↓

\[{e}^{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}\]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (pow E (- (log (fmod (exp x) (sqrt (cos x)))) x)))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

↓

double code(double x) {
	return pow(((double) M_E), (log(fmod(exp(x), sqrt(cos(x)))) - x));
}

Error

Bits error versus x

Derivation

1. Initial program 59.8

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

2. Simplified 59.8

   \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\]
3. Using strategy rm

4. Applied add-exp-log_binary64_798 59.8

   \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}}\]

5. Applied div-exp_binary64_811 59.8

   \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}}\]
6. Using strategy rm

7. Applied *-un-lft-identity_binary64_760 59.8

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

8. Applied exp-prod_binary64_812 59.8

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

9. Simplified 59.8

   \[\leadsto {\color{blue}{e}}^{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}\]

10. Final simplification 59.8

    \[\leadsto {e}^{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))