Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 14.8s

Precision: binary64

Math

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

↓

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

FPCore

(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) (sqrt (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0))))
  (/ (sqrt (sqrt 2.0)) (hypot a2 a1))))

C

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) * sqrt(fma(a1, a1, (a2 * a2)) / sqrt(2.0))) / (sqrt(sqrt(2.0)) / hypot(a2, a1));
}

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. Simplified 0.4

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

3. Applied add-sqr-sqrt_binary64 0.4

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

4. Applied associate-/l*_binary64 0.4

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

5. Simplified 0.4

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

6. Applied associate-*r/_binary64 0.4

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

7. Simplified 0.4

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

8. Applied *-un-lft-identity_binary64 0.4

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

9. Applied add-sqr-sqrt_binary64 0.5

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

10. Applied times-frac_binary64 0.5

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

11. Applied associate-/r*_binary64 0.5

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

12. Taylor expanded in th around inf 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022121 
(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))))