Average Error: 59.8 → 24.8
Time: 12.8s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[e^{-x} \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
e^{-x}

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x) :precision binary64 (exp (- x)))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return exp(-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. Simplified59.8

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

  3. Applied add-exp-log_binary6459.8

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

  4. Applied div-exp_binary6459.8

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

  5. Taylor expanded in x around inf 24.8

    \[\leadsto e^{\color{blue}{-1 \cdot x}} \]

  6. Simplified24.8

    \[\leadsto e^{\color{blue}{-x}} \]

  7. Final simplification24.8

    \[\leadsto e^{-x} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))