Average Error: 0.1 → 0.1
Time: 12.2s
Precision: binary64
\[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
↓
\[\begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ \left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + t_1 \cdot t_1}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \end{array} \]
\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|
↓
\begin{array}{l}
t_1 := \frac{eh}{ew \cdot \tan t}\\
\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + t_1 \cdot t_1}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|
\end{array}

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t)))))
   (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))
↓
(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (fabs
    (+
     (* (* ew (sin t)) (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
     (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))))))
double code(double eh, double ew, double t) {
	return fabs(((ew * sin(t)) * cos(atan((eh / ew) / tan(t)))) + ((eh * cos(t)) * sin(atan((eh / ew) / tan(t)))));
}
↓
double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	return fabs(((ew * sin(t)) * (1.0 / sqrt(1.0 + (t_1 * t_1)))) + ((eh * cos(t)) * sin(atan((eh / ew) / tan(t)))));
}

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

    \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
  2. Using strategy rm

  3. Applied cos-atan_binary640.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2021205 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))