Average Error: 0.5 → 0.5
Time: 13.1s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{\sqrt{2}}\\ t_2 := \sqrt[3]{t_1}\\ \cos th \cdot \left(\frac{1}{t_2 \cdot t_2} \cdot \frac{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{t_1 \cdot t_1}}{t_2}\right) \end{array} \]
\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[3]{\sqrt{2}}\\
t_2 := \sqrt[3]{t_1}\\
\cos th \cdot \left(\frac{1}{t_2 \cdot t_2} \cdot \frac{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{t_1 \cdot t_1}}{t_2}\right)
\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 (cbrt (sqrt 2.0))) (t_2 (cbrt t_1)))
   (*
    (cos th)
    (* (/ 1.0 (* t_2 t_2)) (/ (/ (fma a1 a1 (* a2 a2)) (* t_1 t_1)) t_2)))))
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 = cbrt(sqrt(2.0));
	double t_2 = cbrt(t_1);
	return cos(th) * ((1.0 / (t_2 * t_2)) * ((fma(a1, a1, (a2 * a2)) / (t_1 * t_1)) / t_2));
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

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

  3. Applied add-cube-cbrt_binary640.5

    \[\leadsto \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}} \]

  4. Applied associate-/r*_binary640.5

    \[\leadsto \cos th \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}} \]

  5. Applied add-cube-cbrt_binary640.5

    \[\leadsto \cos th \cdot \frac{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}} \]

  6. Applied *-un-lft-identity_binary640.5

    \[\leadsto \cos th \cdot \frac{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \]

  7. Applied times-frac_binary640.5

    \[\leadsto \cos th \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)} \]

  8. Final simplification0.5

    \[\leadsto \cos th \cdot \left(\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right) \]

Reproduce

herbie shell --seed 2022094 
(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))))