Average Error: 0.5 → 0.4
Time: 10.5s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\left(\sqrt[3]{0.5} \cdot \cos th\right) \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt[3]{\sqrt{2}}} \]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\sqrt[3]{0.5} \cdot \cos th\right) \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt[3]{\sqrt{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
 (* (* (cbrt 0.5) (cos th)) (/ (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 (cbrt(0.5) * cos(th)) * (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. Simplified0.5

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

  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 associate-/r*_binary640.5

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

  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

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

  8. Applied sqrt-prod_binary640.4

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

  9. Applied cbrt-prod_binary640.4

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

  10. Applied times-frac_binary640.4

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

  11. Applied associate-*r*_binary640.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2022081 
(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))))