Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 5.9m

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r28552546 = th;
        double r28552547 = cos(r28552546);
        double r28552548 = 2.0;
        double r28552549 = sqrt(r28552548);
        double r28552550 = r28552547 / r28552549;
        double r28552551 = a1;
        double r28552552 = r28552551 * r28552551;
        double r28552553 = r28552550 * r28552552;
        double r28552554 = a2;
        double r28552555 = r28552554 * r28552554;
        double r28552556 = r28552550 * r28552555;
        double r28552557 = r28552553 + r28552556;
        return r28552557;
}

↓

double f(double a1, double a2, double th) {
        double r28552558 = 1.0;
        double r28552559 = 2.0;
        double r28552560 = sqrt(r28552559);
        double r28552561 = sqrt(r28552560);
        double r28552562 = sqrt(r28552561);
        double r28552563 = r28552558 / r28552562;
        double r28552564 = r28552563 / r28552562;
        double r28552565 = a2;
        double r28552566 = th;
        double r28552567 = cos(r28552566);
        double r28552568 = r28552567 / r28552561;
        double r28552569 = r28552568 * r28552565;
        double r28552570 = r28552565 * r28552569;
        double r28552571 = r28552564 * r28552570;
        double r28552572 = a1;
        double r28552573 = r28552572 * r28552567;
        double r28552574 = r28552560 / r28552572;
        double r28552575 = r28552573 / r28552574;
        double r28552576 = r28552571 + r28552575;
        return r28552576;
}

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. Taylor expanded around -inf 0.5

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

3. Simplified 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied add-sqr-sqrt 0.5

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

11. Applied sqrt-prod 0.5

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

12. Applied *-un-lft-identity 0.5

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

13. Applied times-frac 0.5

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

14. Applied times-frac 0.5

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

15. Applied associate-*l* 0.5

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

16. Simplified 0.4

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

18. Applied associate-*l* 0.4

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

19. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))