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

Average Error: 15.3 → 0.3

Time: 13.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}\]

C

double f(double r, double a, double b) {
        double r16574 = r;
        double r16575 = b;
        double r16576 = sin(r16575);
        double r16577 = r16574 * r16576;
        double r16578 = a;
        double r16579 = r16578 + r16575;
        double r16580 = cos(r16579);
        double r16581 = r16577 / r16580;
        return r16581;
}

↓

double f(double r, double a, double b) {
        double r16582 = r;
        double r16583 = b;
        double r16584 = sin(r16583);
        double r16585 = a;
        double r16586 = sin(r16585);
        double r16587 = -r16586;
        double r16588 = cos(r16585);
        double r16589 = cos(r16583);
        double r16590 = r16588 * r16589;
        double r16591 = fma(r16584, r16587, r16590);
        double r16592 = r16584 / r16591;
        double r16593 = r16582 * r16592;
        return r16593;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.3

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

3. Applied cos-sum 0.3

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

5. Applied fma-neg 0.3

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

6. Simplified 0.3

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

8. Applied *-un-lft-identity 0.3

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

9. Applied times-frac 0.3

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

10. Simplified 0.3

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

11. Taylor expanded around inf 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \cos a \cdot \cos b\right)}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))