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

Average Error: 15.1 → 0.3

Time: 6.5s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r17138 = r;
        double r17139 = b;
        double r17140 = sin(r17139);
        double r17141 = a;
        double r17142 = r17141 + r17139;
        double r17143 = cos(r17142);
        double r17144 = r17140 / r17143;
        double r17145 = r17138 * r17144;
        return r17145;
}

↓

double f(double r, double a, double b) {
        double r17146 = r;
        double r17147 = b;
        double r17148 = sin(r17147);
        double r17149 = r17146 * r17148;
        double r17150 = a;
        double r17151 = cos(r17150);
        double r17152 = cos(r17147);
        double r17153 = sin(r17150);
        double r17154 = r17153 * r17148;
        double r17155 = -r17154;
        double r17156 = fma(r17151, r17152, r17155);
        double r17157 = r17149 / r17156;
        return r17157;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

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

3. Applied cos-sum 0.3

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

5. Applied fma-neg 0.3

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

7. Applied associate-*r/ 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))