Average Error: 59.5 → 59.6
Time: 13.4s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \left(e^{\sqrt[3]{\sqrt[3]{\cos x}}}\right) \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \left(e^{\sqrt[3]{\sqrt[3]{\cos x}}}\right) \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}}

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (/
  (fmod
   (exp x)
   (*
    (*
     (log (exp (cbrt (cbrt (cos x)))))
     (pow (cbrt (pow (cos x) 0.16666666666666666)) 4.0))
    (expm1 (log1p (cbrt (sqrt (cos x)))))))
  (exp x)))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return fmod(exp(x), ((log(exp(cbrt(cbrt(cos(x))))) * pow(cbrt(pow(cos(x), 0.16666666666666666)), 4.0)) * expm1(log1p(cbrt(sqrt(cos(x))))))) / exp(x);
}

Error

Bits error versus x

Derivation

  1. Initial program 59.5

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

  2. Simplified59.5

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

  3. Applied add-cube-cbrt_binary6459.6

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

  4. Applied expm1-log1p-u_binary6459.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\sqrt[3]{\sqrt{\cos x}} \cdot \sqrt[3]{\sqrt{\cos x}}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)}\right)\right)}{e^{x}} \]

  5. Applied add-cube-cbrt_binary6459.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\sqrt[3]{\sqrt{\color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}}}} \cdot \sqrt[3]{\sqrt{\cos x}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

  6. Applied sqrt-prod_binary6459.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\sqrt[3]{\color{blue}{\sqrt{\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}} \cdot \sqrt{\sqrt[3]{\cos x}}}} \cdot \sqrt[3]{\sqrt{\cos x}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

  7. Applied cbrt-prod_binary6459.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{\cos x}}}\right)} \cdot \sqrt[3]{\sqrt{\cos x}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

  8. Applied associate-*l*_binary6459.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{\cos x}}} \cdot \sqrt[3]{\sqrt{\cos x}}\right)\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

  9. Simplified59.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\sqrt[3]{\sqrt{\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}}} \cdot \color{blue}{{\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

  10. Applied add-log-exp_binary6459.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\color{blue}{\log \left(e^{\sqrt[3]{\sqrt{\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}}}}\right)} \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

  11. Simplified59.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \color{blue}{\left(e^{\sqrt[3]{\sqrt[3]{\cos x}}}\right)} \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

  12. Final simplification59.6

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\log \left(e^{\sqrt[3]{\sqrt[3]{\cos x}}}\right) \cdot {\left(\sqrt[3]{{\cos x}^{0.16666666666666666}}\right)}^{4}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\cos x}}\right)\right)\right)\right)}{e^{x}} \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))