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

Average Error: 15.1 → 0.4

Time: 34.2s

Precision: 64

Math

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

↓

\[r \cdot \left(\frac{\sin b}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}} \cdot \frac{1}{\cos a \cdot \cos b + \sin a \cdot \sin b}\right)\]

C

double f(double r, double a, double b) {
        double r932567 = r;
        double r932568 = b;
        double r932569 = sin(r932568);
        double r932570 = a;
        double r932571 = r932570 + r932568;
        double r932572 = cos(r932571);
        double r932573 = r932569 / r932572;
        double r932574 = r932567 * r932573;
        return r932574;
}

↓

double f(double r, double a, double b) {
        double r932575 = r;
        double r932576 = b;
        double r932577 = sin(r932576);
        double r932578 = a;
        double r932579 = cos(r932578);
        double r932580 = cos(r932576);
        double r932581 = r932579 * r932580;
        double r932582 = sin(r932578);
        double r932583 = r932582 * r932577;
        double r932584 = r932581 - r932583;
        double r932585 = r932581 + r932583;
        double r932586 = r932584 / r932585;
        double r932587 = r932577 / r932586;
        double r932588 = 1.0;
        double r932589 = r932588 / r932585;
        double r932590 = r932587 * r932589;
        double r932591 = r932575 * r932590;
        return r932591;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
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 flip-- 0.4

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.4

   \[\leadsto r \cdot \frac{\sin b}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}}}\]

8. Applied difference-of-squares 0.3

   \[\leadsto r \cdot \frac{\sin b}{\frac{\color{blue}{\left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}{1 \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}}\]

9. Applied times-frac 0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{1} \cdot \frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

10. Applied *-un-lft-identity 0.3

    \[\leadsto r \cdot \frac{\color{blue}{1 \cdot \sin b}}{\frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{1} \cdot \frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}\]

11. Applied times-frac 0.4

    \[\leadsto r \cdot \color{blue}{\left(\frac{1}{\frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{1}} \cdot \frac{\sin b}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}\right)}\]

12. Final simplification 0.4

    \[\leadsto r \cdot \left(\frac{\sin b}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}} \cdot \frac{1}{\cos a \cdot \cos b + \sin a \cdot \sin b}\right)\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))