Average Error: 15.3 → 0.4
Time: 23.7s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\log \left(e^{\sin a \cdot \sin b}\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\log \left(e^{\sin a \cdot \sin b}\right)\right)}
double f(double r, double a, double b) {
        double r24570 = r;
        double r24571 = b;
        double r24572 = sin(r24571);
        double r24573 = r24570 * r24572;
        double r24574 = a;
        double r24575 = r24574 + r24571;
        double r24576 = cos(r24575);
        double r24577 = r24573 / r24576;
        return r24577;
}
↓
double f(double r, double a, double b) {
        double r24578 = r;
        double r24579 = b;
        double r24580 = sin(r24579);
        double r24581 = a;
        double r24582 = cos(r24581);
        double r24583 = cos(r24579);
        double r24584 = sin(r24581);
        double r24585 = r24584 * r24580;
        double r24586 = exp(r24585);
        double r24587 = log(r24586);
        double r24588 = -r24587;
        double r24589 = fma(r24582, r24583, r24588);
        double r24590 = r24580 / r24589;
        double r24591 = r24578 * r24590;
        return r24591;
}

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-sum0.3

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

  5. Applied fma-neg0.3

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

  7. Applied *-un-lft-identity0.3

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  11. Applied add-log-exp0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))