Average Error: 0.5 → 0.5
Time: 12.0s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\left(\cos th \cdot \left(\mathsf{hypot}\left(a2, a1\right) \cdot \sqrt[3]{0.5}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt[3]{\sqrt{2}}} \]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\cos th \cdot \left(\mathsf{hypot}\left(a2, a1\right) \cdot \sqrt[3]{0.5}\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\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
 (*
  (* (cos th) (* (hypot a2 a1) (cbrt 0.5)))
  (/ (sqrt (fma a1 a1 (* a2 a2))) (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) * (hypot(a2, a1) * cbrt(0.5))) * (sqrt(fma(a1, a1, (a2 * a2))) / 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 add-sqr-sqrt_binary640.5

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

  5. Applied times-frac_binary640.5

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

  6. Applied associate-*r*_binary640.5

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

  7. Simplified0.5

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

  8. Taylor expanded in th around inf 0.7

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2022087 
(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))))