Average Error: 0.5 → 0.5
Time: 3.5min
Precision: binary64
Cost: 1536
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \sqrt[3]{0.5} \cdot \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \left(a2 \cdot a2\right)\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \sqrt[3]{0.5} \cdot \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \left(a2 \cdot a2\right)\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) (* a1 a1)) (sqrt 2.0))
  (* (cbrt 0.5) (* (/ (cos th) (fabs (cbrt (sqrt 2.0)))) (* a2 a2)))))
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) * (a1 * a1)) / sqrt(2.0)) + (cbrt(0.5) * ((cos(th) / fabs(cbrt(sqrt(2.0)))) * (a2 * a2)));
}

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
Alternative 1
Accuracy0.5
Cost2624
\[\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \left(\frac{1}{\sqrt{\sqrt[3]{\left|\sqrt[3]{2}\right|}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt[3]{2}}}}}\right)\right) \cdot \left(a2 \cdot a2\right)\]
Alternative 2
Accuracy0.5
Cost1664
\[\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \frac{\cos th \cdot \sqrt{\frac{1}{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a2 \cdot a2\right)\]
Alternative 3
Accuracy1.4
Cost3712
\[\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \left(\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left|\sqrt[3]{\sqrt{2}}\right|}}{\sqrt[3]{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{\sqrt{2}}}}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{\sqrt[3]{\sqrt{2}}}}}{\sqrt[3]{\sqrt{\sqrt[3]{\sqrt{2}}}}}\right)\right) \cdot \left(a2 \cdot a2\right)\]

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. Using strategy rm

  3. Applied add-sqr-sqrt_binary64_1000.5

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

  4. Applied associate-/r*_binary64_220.5

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

  6. Applied associate-*l/_binary64_210.5

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

  8. Applied add-cube-cbrt_binary64_1130.5

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

  9. Applied sqrt-prod_binary64_940.6

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

  10. Applied div-inv_binary64_750.5

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

  11. Applied times-frac_binary64_840.5

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

  12. Simplified0.5

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

  13. Taylor expanded around inf 0.6

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020322 
(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))))