Average Error: 0.5 → 0.4
Time: 10.5s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{2}}} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{2}}} \cdot \cos th \]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\sqrt{\frac{1}{\sqrt{2}}} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{2}}} \cdot \cos th

(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
 (*
  (/ (* (sqrt (/ 1.0 (sqrt 2.0))) (fma a1 a1 (* a2 a2))) (sqrt (sqrt 2.0)))
  (cos th)))
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 ((sqrt(1.0 / sqrt(2.0)) * fma(a1, a1, (a2 * a2))) / sqrt(sqrt(2.0))) * cos(th);
}

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 *-commutative_binary640.5

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

  4. Applied add-sqr-sqrt_binary640.6

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

  5. Applied associate-/r*_binary640.5

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

  6. Simplified0.5

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

  7. Taylor expanded in a2 around 0 0.4

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{\sqrt{2}}} \cdot {a2}^{2} + {a1}^{2} \cdot \sqrt{\frac{1}{\sqrt{2}}}}}{\sqrt{\sqrt{2}}} \cdot \cos th \]

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2021329 
(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))))