Average Error: 59.6 → 59.6
Time: 9.0s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}\]
↓
\[\frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}} \cdot \log \left(e^{\frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}}}\right)\]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}} \cdot \log \left(e^{\frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}}}\right)
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (*
  (/
   (sqrt (fmod (exp x) (* (fabs (cbrt (cos x))) (sqrt (cbrt (cos x))))))
   (sqrt (exp x)))
  (log
   (exp
    (/
     (sqrt (fmod (exp x) (* (fabs (cbrt (cos x))) (sqrt (cbrt (cos x))))))
     (sqrt (exp x)))))))
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) ((((double) sqrt(((double) fmod(((double) exp(x)), ((double) (((double) fabs(((double) cbrt(((double) cos(x)))))) * ((double) sqrt(((double) cbrt(((double) cos(x)))))))))))) / ((double) sqrt(((double) exp(x))))) * ((double) log(((double) exp((((double) sqrt(((double) fmod(((double) exp(x)), ((double) (((double) fabs(((double) cbrt(((double) cos(x)))))) * ((double) sqrt(((double) cbrt(((double) cos(x)))))))))))) / ((double) sqrt(((double) exp(x)))))))))));
}

Error

Bits error versus x

Derivation

  1. Initial program 59.6

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

  2. Simplified59.6

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

  4. Applied add-cube-cbrt_binary6459.6

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

  5. Applied sqrt-prod_binary6459.6

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

  6. Simplified59.6

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

  8. Applied add-sqr-sqrt_binary6459.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}}\]

  9. Applied add-sqr-sqrt_binary6459.6

    \[\leadsto \frac{\color{blue}{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)} \cdot \sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}\]

  10. Applied times-frac_binary6459.6

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

  12. Applied add-log-exp_binary6459.6

    \[\leadsto \frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}} \cdot \color{blue}{\log \left(e^{\frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}}}\right)}\]

  13. Final simplification59.6

    \[\leadsto \frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}} \cdot \log \left(e^{\frac{\sqrt{\left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}{\sqrt{e^{x}}}}\right)\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))