Average Error: 0.5 → 0.3
Time: 13.8s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\left(\cos th \cdot \mathsf{hypot}\left(a2, a1\right)\right) \cdot \sqrt{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{2}} \]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\cos th \cdot \mathsf{hypot}\left(a2, a1\right)\right) \cdot \sqrt{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{2}}

(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
 (* (* (cos th) (hypot a2 a1)) (sqrt (/ (fma a1 a1 (* a2 a2)) 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) {
	return (cos(th) * hypot(a2, a1)) * sqrt(fma(a1, a1, (a2 * a2)) / 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 *-un-lft-identity_binary640.5

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

  4. Applied sqrt-prod_binary640.5

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

  5. Applied add-sqr-sqrt_binary640.5

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

  6. Applied times-frac_binary640.5

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

  7. Applied associate-*r*_binary640.5

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

  8. Simplified0.5

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

  9. Applied sqrt-undiv_binary640.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2021225 
(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))))