Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 27.7s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r72655 = th;
        double r72656 = cos(r72655);
        double r72657 = 2.0;
        double r72658 = sqrt(r72657);
        double r72659 = r72656 / r72658;
        double r72660 = a1;
        double r72661 = r72660 * r72660;
        double r72662 = r72659 * r72661;
        double r72663 = a2;
        double r72664 = r72663 * r72663;
        double r72665 = r72659 * r72664;
        double r72666 = r72662 + r72665;
        return r72666;
}

↓

double f(double a1, double a2, double th) {
        double r72667 = 1.0;
        double r72668 = 2.0;
        double r72669 = cbrt(r72668);
        double r72670 = fabs(r72669);
        double r72671 = sqrt(r72670);
        double r72672 = r72667 / r72671;
        double r72673 = th;
        double r72674 = cos(r72673);
        double r72675 = sqrt(r72668);
        double r72676 = sqrt(r72675);
        double r72677 = r72674 / r72676;
        double r72678 = sqrt(r72669);
        double r72679 = sqrt(r72678);
        double r72680 = r72677 / r72679;
        double r72681 = r72672 * r72680;
        double r72682 = a1;
        double r72683 = r72682 * r72682;
        double r72684 = r72681 * r72683;
        double r72685 = r72677 / r72676;
        double r72686 = a2;
        double r72687 = r72686 * r72686;
        double r72688 = r72685 * r72687;
        double r72689 = r72684 + r72688;
        return r72689;
}

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. Using strategy rm

3. Applied add-sqr-sqrt 0.5

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

4. Applied sqrt-prod 0.5

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

5. Applied associate-/r* 0.5

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied sqrt-prod 0.5

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

9. Applied associate-/r* 0.5

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

11. Applied *-un-lft-identity 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied add-cube-cbrt 0.5

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

15. Applied sqrt-prod 0.5

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

16. Applied sqrt-prod 0.5

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

17. Applied *-un-lft-identity 0.5

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

18. Applied times-frac 0.5

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

19. Applied times-frac 0.5

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

20. Simplified 0.5

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

21. Simplified 0.5

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

22. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019303 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))