Average Error: 59.7 → 47.8
Time: 12.7s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\log \left(e^{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{1 + x}}\right) \]
\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\log \left(e^{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{1 + x}}\right)

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (log (exp (/ (fmod (+ 1.0 x) (sqrt (cos x))) (+ 1.0 x)))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return log(exp(fmod((1.0 + x), sqrt(cos(x))) / (1.0 + 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. Simplified59.7

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

  3. Taylor expanded in x around 0 47.6

    \[\leadsto \frac{\left(\color{blue}{\left(1 + x\right)} \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]

  4. Taylor expanded in x around 0 59.4

    \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{\color{blue}{1 + x}} \]

  5. Simplified59.4

    \[\leadsto \frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{\color{blue}{x + 1}} \]

  6. Applied add-log-exp_binary6447.8

    \[\leadsto \color{blue}{\log \left(e^{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{x + 1}}\right)} \]

  7. Final simplification47.8

    \[\leadsto \log \left(e^{\frac{\left(\left(1 + x\right) \bmod \left(\sqrt{\cos x}\right)\right)}{1 + x}}\right) \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))