Average Error: 0.5 → 0.5
Time: 10.5s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{\sqrt{2}}\\ \frac{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{t_1} \cdot \cos th}{t_1 \cdot t_1} \end{array} \]
\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{\frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{t_1} \cdot \cos th}{t_1 \cdot t_1}
\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))))
   (/ (* (/ (fma a2 a2 (* a1 a1)) t_1) (cos th)) (* t_1 t_1))))
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 ((fma(a2, a2, (a1 * a1)) / t_1) * cos(th)) / (t_1 * t_1);
}

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 add-cube-cbrt_binary641.3

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

  5. Applied times-frac_binary641.3

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

  6. Applied associate-*l/_binary641.3

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

  7. Simplified0.5

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

  8. Applied *-commutative_binary640.5

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

  9. Applied associate-*l/_binary640.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2021215 
(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))))