Average Error: 15.5 → 0.4
Time: 53.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[-\frac{r \cdot \sin b}{\mathsf{fma}\left(-\cos b, \cos a, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
-\frac{r \cdot \sin b}{\mathsf{fma}\left(-\cos b, \cos a, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin b \cdot \sin a\right)\right)\right)}
double f(double r, double a, double b) {
        double r926128 = r;
        double r926129 = b;
        double r926130 = sin(r926129);
        double r926131 = a;
        double r926132 = r926131 + r926129;
        double r926133 = cos(r926132);
        double r926134 = r926130 / r926133;
        double r926135 = r926128 * r926134;
        return r926135;
}
↓
double f(double r, double a, double b) {
        double r926136 = r;
        double r926137 = b;
        double r926138 = sin(r926137);
        double r926139 = r926136 * r926138;
        double r926140 = cos(r926137);
        double r926141 = -r926140;
        double r926142 = a;
        double r926143 = cos(r926142);
        double r926144 = sin(r926142);
        double r926145 = r926138 * r926144;
        double r926146 = expm1(r926145);
        double r926147 = log1p(r926146);
        double r926148 = fma(r926141, r926143, r926147);
        double r926149 = r926139 / r926148;
        double r926150 = -r926149;
        return r926150;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.5

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

  3. Applied cos-sum0.3

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

  5. Applied associate-*r/0.3

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

  7. Applied frac-2neg0.3

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

  8. Simplified0.3

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

  10. Applied log1p-expm1-u0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))