Average Error: 0.5 → 0.4
Time: 11.1s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\cos th \cdot \left(\sqrt[3]{0.5} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt[3]{\sqrt{2}}}\right) \]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\cos th \cdot \left(\sqrt[3]{0.5} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt[3]{\sqrt{2}}}\right)

(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) (* (cbrt 0.5) (/ (fma a2 a2 (* a1 a1)) (cbrt (sqrt 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) * (cbrt(0.5) * (fma(a2, a2, (a1 * a1)) / cbrt(sqrt(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. Simplified0.4

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

  3. Applied add-cube-cbrt_binary640.4

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

  4. Applied associate-/r*_binary640.4

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

  5. Taylor expanded in a1 around 0 0.6

    \[\leadsto \cos th \cdot \frac{\color{blue}{{\left(\frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}^{0.3333333333333333} \cdot {a2}^{2} + {a1}^{2} \cdot {\left(\frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}^{0.3333333333333333}}}{\sqrt[3]{\sqrt{2}}} \]

  6. Simplified0.4

    \[\leadsto \cos th \cdot \frac{\color{blue}{\sqrt[3]{0.5} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}}{\sqrt[3]{\sqrt{2}}} \]

  7. Applied *-un-lft-identity_binary640.4

    \[\leadsto \cos th \cdot \frac{\sqrt[3]{0.5} \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\color{blue}{1 \cdot \sqrt[3]{\sqrt{2}}}} \]

  8. Applied times-frac_binary640.4

    \[\leadsto \cos th \cdot \color{blue}{\left(\frac{\sqrt[3]{0.5}}{1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt[3]{\sqrt{2}}}\right)} \]

  9. Final simplification0.4

    \[\leadsto \cos th \cdot \left(\sqrt[3]{0.5} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt[3]{\sqrt{2}}}\right) \]

Reproduce

herbie shell --seed 2022095 
(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))))