Average Error: 0.1 → 0.1
Time: 13.8s
Precision: binary64
\[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
↓
\[\begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\mathsf{fma}\left(ew \cdot \sin t, \mathsf{expm1}\left(\mathsf{log1p}\left(\cos t_1\right)\right), \left(eh \cdot \cos t\right) \cdot \sin t_1\right)\right| \end{array} \]
\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)\\
\left|\mathsf{fma}\left(ew \cdot \sin t, \mathsf{expm1}\left(\mathsf{log1p}\left(\cos t_1\right)\right), \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)))))
   (fabs
    (fma
     (* ew (sin t))
     (expm1 (log1p (cos t_1)))
     (* (* 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));
	return fabs(fma((ew * sin(t)), expm1(log1p(cos(t_1))), ((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 expm1-log1p-u_binary640.1

    \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right)}, \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right| \]

  4. Final simplification0.1

    \[\leadsto \left|\mathsf{fma}\left(ew \cdot \sin t, \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right), \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)\right| \]

Reproduce

herbie shell --seed 2022067 
(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))))))))