expfmod

Average Error: 59.5 → 59.5

Time: 9.0s

Precision: binary64

• could not determine a ground truth for program body

  1. x = -1.5611167895055562e+201

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return (((double) (((double) cbrt(((double) fmod(((double) exp(x)), ((double) cbrt(((double) pow(((double) sqrt(((double) cos(x)))), 3.0)))))))) * ((double) cbrt(((double) fmod(((double) exp(x)), ((double) cbrt(((double) pow(((double) sqrt(((double) cos(x)))), 3.0)))))))))) / ((double) cbrt(((double) exp(((double) (((double) (x * 3.0)) - ((double) log(((double) fmod(((double) exp(x)), ((double) sqrt(((double) cos(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. Simplified 59.5

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

4. Applied add-cbrt-cube 59.5

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

5. Simplified 59.5

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

7. Applied add-cube-cbrt 59.5

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

8. Applied associate-/l* 59.5

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

9. Simplified 59.5

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

11. Applied add-cbrt-cube 59.5

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

12. Applied cbrt-undiv 59.5

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

13. Simplified 59.5

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

15. Applied add-exp-log 59.5

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

16. Applied pow-exp 59.5

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

17. Applied div-exp 59.5

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

18. Final simplification 59.5

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))