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

Average Error: 14.7 → 0.4

Time: 23.6s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r25137 = r;
        double r25138 = b;
        double r25139 = sin(r25138);
        double r25140 = r25137 * r25139;
        double r25141 = a;
        double r25142 = r25141 + r25138;
        double r25143 = cos(r25142);
        double r25144 = r25140 / r25143;
        return r25144;
}

↓

double f(double r, double a, double b) {
        double r25145 = b;
        double r25146 = sin(r25145);
        double r25147 = a;
        double r25148 = sin(r25147);
        double r25149 = cos(r25147);
        double r25150 = -r25149;
        double r25151 = cos(r25145);
        double r25152 = r25150 * r25151;
        double r25153 = fma(r25148, r25146, r25152);
        double r25154 = r;
        double r25155 = r25153 / r25154;
        double r25156 = r25146 / r25155;
        double r25157 = -r25156;
        return r25157;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.7

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 frac-2neg 0.3

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

6. Simplified 0.3

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

7. Taylor expanded around inf 0.3

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

8. Simplified 0.3

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

10. Applied neg-sub0 0.3

    \[\leadsto \frac{\color{blue}{0 - r \cdot \sin b}}{\mathsf{fma}\left(\sin a, \sin b, \left(-\cos a\right) \cdot \cos b\right)}\]

11. Applied div-sub 0.3

    \[\leadsto \color{blue}{\frac{0}{\mathsf{fma}\left(\sin a, \sin b, \left(-\cos a\right) \cdot \cos b\right)} - \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \sin b, \left(-\cos a\right) \cdot \cos b\right)}}\]

12. Simplified 0.3

    \[\leadsto \color{blue}{0} - \frac{r \cdot \sin b}{\mathsf{fma}\left(\sin a, \sin b, \left(-\cos a\right) \cdot \cos b\right)}\]

13. Simplified 0.4

    \[\leadsto 0 - \color{blue}{\frac{\sin b}{\frac{\mathsf{fma}\left(\sin a, \sin b, \left(-\cos a\right) \cdot \cos b\right)}{r}}}\]

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))