Average Error: 0.5 → 0.5
Time: 9.7s
Precision: binary64
\[[a1, a2] = \mathsf{sort}([a1, a2]) \\]
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{\sqrt{2}}\\ \frac{\cos th}{t_1} \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{{t_1}^{2}} \end{array} \]
\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[3]{\sqrt{2}}\\
\frac{\cos th}{t_1} \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{{t_1}^{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 (cbrt (sqrt 2.0))))
   (* (/ (cos th) t_1) (/ (fma a1 a1 (* a2 a2)) (pow t_1 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 = cbrt(sqrt(2.0));
	return (cos(th) / t_1) * (fma(a1, a1, (a2 * a2)) / pow(t_1, 2.0));
}

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 associate-*r/_binary640.5

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

  4. Applied add-cube-cbrt_binary640.5

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

  5. Applied associate-/r*_binary640.5

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

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

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

  7. Applied sqrt-prod_binary640.5

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

  8. Applied cbrt-prod_binary640.5

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

  9. Applied times-frac_binary640.6

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

  10. Applied times-frac_binary640.6

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

  11. Simplified0.6

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2022076 
(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))))