Average Error: 0.1 → 0.1
Time: 12.2s
Precision: binary64
\[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
↓
\[\left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right| \]
\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|
↓
\left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right|

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew))))
   (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))
↓
(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* ew (* (cos t) (cos (atan (/ (* (- eh) (tan t)) ew)))))
   (* (* eh (sin t)) (sin (atan (* (- eh) (/ (tan t) ew))))))))
double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) * cos(atan((-eh * tan(t)) / ew))) - ((eh * sin(t)) * sin(atan((-eh * tan(t)) / ew))));
}
↓
double code(double eh, double ew, double t) {
	return fabs((ew * (cos(t) * cos(atan((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(-eh * (tan(t) / ew)))));
}

Error

Bits error versus eh

Bits error versus ew

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

  2. Applied associate-*l*_binary640.1

    \[\leadsto \left|\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

  3. Applied *-un-lft-identity_binary640.1

    \[\leadsto \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{\color{blue}{1 \cdot ew}}\right)\right| \]

  4. Applied times-frac_binary640.1

    \[\leadsto \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-eh}{1} \cdot \frac{\tan t}{ew}\right)}\right| \]

  5. Simplified0.1

    \[\leadsto \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\color{blue}{\left(-eh\right)} \cdot \frac{\tan t}{ew}\right)\right| \]

  6. Final simplification0.1

    \[\leadsto \left|ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right| \]

Reproduce

herbie shell --seed 2022082 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))