Average Error: 59.5 → 59.5
Time: 11.8s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (exp (- (log (fmod (exp x) (sqrt (cos x)))) x)))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return exp(log(fmod(exp(x), sqrt(cos(x)))) - 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_binary6459.5

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

  4. Applied sqrt-prod_binary6459.5

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

  5. Simplified59.5

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

  6. Applied add-exp-log_binary6459.5

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

  7. Applied div-exp_binary6459.5

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

  8. Simplified59.5

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

  9. Final simplification59.5

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

Reproduce

herbie shell --seed 2021280 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))