expfmod

Average Error: 59.6 → 59.6

Time: 10.8s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (cbrt (pow (sqrt (cos x)) 3.0))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

↓

double code(double x) {
	return fmod(exp(x), cbrt(pow(sqrt(cos(x)), 3.0))) / 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. Simplified 59.6

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

4. Applied add-cbrt-cube_binary64_784 59.6

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

5. Simplified 59.6

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

6. Final simplification 59.6

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

Reproduce

herbie shell --seed 2020281 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))