Average Error: 0.3 → 0.3
Time: 11.7s
Precision: binary64
\[[a1, a2] = \mathsf{sort}([a1, a2]) \\]
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}\\ \left(\cos th \cdot t_1\right) \cdot \frac{t_1}{\sqrt{2}} \end{array} \]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\begin{array}{l}
t_1 := \sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}\\
\left(\cos th \cdot t_1\right) \cdot \frac{t_1}{\sqrt{2}}
\end{array}

(FPCore (a1 a2 th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1 a1))
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))
↓
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (sqrt (fma a1 a1 (* a2 a2)))))
   (* (* (cos th) t_1) (/ t_1 (sqrt 2.0)))))
double code(double a1, double a2, double th) {
	return ((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
}
↓
double code(double a1, double a2, double th) {
	double t_1 = sqrt(fma(a1, a1, (a2 * a2)));
	return (cos(th) * t_1) * (t_1 / sqrt(2.0));
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.3

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

  2. Simplified0.3

    \[\leadsto \color{blue}{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}} \]

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

    \[\leadsto \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{1 \cdot \sqrt{2}}} \]

  4. Applied add-sqr-sqrt_binary640.3

    \[\leadsto \cos th \cdot \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \cdot \sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}}{1 \cdot \sqrt{2}} \]

  5. Applied times-frac_binary640.3

    \[\leadsto \cos th \cdot \color{blue}{\left(\frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}}\right)} \]

  6. Applied associate-*r*_binary640.3

    \[\leadsto \color{blue}{\left(\cos th \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{1}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}}} \]

  7. Final simplification0.3

    \[\leadsto \left(\cos th \cdot \sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}\right) \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))