Average Error: 0.5 → 0.5
Time: 8.2s
Precision: binary64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th\right) \cdot \frac{\sqrt[3]{0.5}}{\sqrt[3]{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th\right) \cdot \frac{\sqrt[3]{0.5}}{\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
 (* (* (+ (* a1 a1) (* a2 a2)) (cos th)) (/ (cbrt 0.5) (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 (((a1 * a1) + (a2 * a2)) * cos(th)) * (cbrt(0.5) / cbrt(sqrt(2.0)));
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

    \[\leadsto \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt_binary640.5

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

  5. Applied associate-/r*_binary640.5

    \[\leadsto \cos th \cdot \color{blue}{\frac{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}}\]

  6. Taylor expanded around 0 0.6

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

  7. Simplified0.4

    \[\leadsto \cos th \cdot \frac{\color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt[3]{0.5}}}{\sqrt[3]{\sqrt{2}}}\]
  8. Using strategy rm

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

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

  10. Applied times-frac_binary640.5

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

  11. Applied associate-*r*_binary640.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020268 
(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))))