expfmod

Average Error: 59.7 → 59.7

Time: 8.5s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 59.7

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

2. Simplified 59.7

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

4. Applied add-exp-log_binary64 59.7

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

5. Applied div-exp_binary64 59.7

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

7. Applied add-log-exp_binary64 59.7

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

8. Final simplification 59.7

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

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))