r*sin(b)/cos(a+b), B

Average Error: 15.2 → 0.5

Time: 7.1s

Precision: 64

Math

\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]

↓

\[\left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\]

C

double f(double r, double a, double b) {
        double r17435 = r;
        double r17436 = b;
        double r17437 = sin(r17436);
        double r17438 = a;
        double r17439 = r17438 + r17436;
        double r17440 = cos(r17439);
        double r17441 = r17437 / r17440;
        double r17442 = r17435 * r17441;
        return r17442;
}

↓

double f(double r, double a, double b) {
        double r17443 = r;
        double r17444 = b;
        double r17445 = sin(r17444);
        double r17446 = a;
        double r17447 = cos(r17446);
        double r17448 = cos(r17444);
        double r17449 = r17447 * r17448;
        double r17450 = 3.0;
        double r17451 = pow(r17449, r17450);
        double r17452 = sin(r17446);
        double r17453 = r17452 * r17445;
        double r17454 = pow(r17453, r17450);
        double r17455 = r17451 - r17454;
        double r17456 = r17445 / r17455;
        double r17457 = r17443 * r17456;
        double r17458 = r17449 * r17449;
        double r17459 = r17453 * r17453;
        double r17460 = r17449 * r17453;
        double r17461 = r17459 + r17460;
        double r17462 = r17458 + r17461;
        double r17463 = r17457 * r17462;
        return r17463;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.2

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
2. Using strategy rm

3. Applied cos-sum 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
4. Using strategy rm

5. Applied flip3-- 0.4

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}}\]

6. Applied associate-/r/ 0.4

   \[\leadsto r \cdot \color{blue}{\left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}} \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\right)}\]

7. Applied associate-*r* 0.5

   \[\leadsto \color{blue}{\left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)}\]

8. Final simplification 0.5

   \[\leadsto \left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\]

Reproduce

herbie shell --seed 2019344 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))