Average Error: 0.1 → 0.1
Time: 20.1s
Precision: binary64
\[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
↓
\[\begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_2 := \sqrt{\cos t_1}\\ \left|\mathsf{fma}\left(ew \cdot \sin t, t_2 \cdot t_2, \left(eh \cdot \cos t\right) \cdot \sin t_1\right)\right| \end{array} \]
\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 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\
t_2 := \sqrt{\cos t_1}\\
\left|\mathsf{fma}\left(ew \cdot \sin t, t_2 \cdot t_2, \left(eh \cdot \cos t\right) \cdot \sin t_1\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 (atan (/ (/ eh ew) (tan t)))) (t_2 (sqrt (cos t_1))))
   (fabs (fma (* ew (sin t)) (* t_2 t_2) (* (* eh (cos t)) (sin t_1))))))
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 = atan((eh / ew) / tan(t));
	double t_2 = sqrt(cos(t_1));
	return fabs(fma((ew * sin(t)), (t_2 * t_2), ((eh * cos(t)) * sin(t_1))));
}

Error

Bits error versus eh

Bits error versus ew

Bits error versus t

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. Simplified0.1

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

  3. Applied add-sqr-sqrt_binary640.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022068 
(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))))))))