Average Error: 15.3 → 0.3
Time: 25.9s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{-\sin b}{\mathsf{fma}\left(\cos a, -\cos b, \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)\right)}\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{-\sin b}{\mathsf{fma}\left(\cos a, -\cos b, \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)\right)}
double f(double r, double a, double b) {
        double r25793 = r;
        double r25794 = b;
        double r25795 = sin(r25794);
        double r25796 = r25793 * r25795;
        double r25797 = a;
        double r25798 = r25797 + r25794;
        double r25799 = cos(r25798);
        double r25800 = r25796 / r25799;
        return r25800;
}
↓
double f(double r, double a, double b) {
        double r25801 = r;
        double r25802 = b;
        double r25803 = sin(r25802);
        double r25804 = -r25803;
        double r25805 = a;
        double r25806 = cos(r25805);
        double r25807 = cos(r25802);
        double r25808 = -r25807;
        double r25809 = sin(r25805);
        double r25810 = r25803 * r25809;
        double r25811 = log1p(r25810);
        double r25812 = expm1(r25811);
        double r25813 = fma(r25806, r25808, r25812);
        double r25814 = r25804 / r25813;
        double r25815 = r25801 * r25814;
        return r25815;
}

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 frac-2neg0.3

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

  6. Simplified0.3

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

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

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

  9. Applied distribute-rgt-neg-in0.3

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  13. Applied expm1-log1p-u0.3

    \[\leadsto r \cdot \frac{-\sin b}{\mathsf{fma}\left(\cos a, -\cos b, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)}\right)}\]

  14. Final simplification0.3

    \[\leadsto r \cdot \frac{-\sin b}{\mathsf{fma}\left(\cos a, -\cos b, \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)\right)}\]

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))))