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

Average Error: 15.3 → 0.4

Time: 7.3s

Precision: 64

Math

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

↓

\[r \cdot \left(\sin b \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}\right)\]

C

double f(double r, double a, double b) {
        double r17927 = r;
        double r17928 = b;
        double r17929 = sin(r17928);
        double r17930 = a;
        double r17931 = r17930 + r17928;
        double r17932 = cos(r17931);
        double r17933 = r17929 / r17932;
        double r17934 = r17927 * r17933;
        return r17934;
}

↓

double f(double r, double a, double b) {
        double r17935 = r;
        double r17936 = b;
        double r17937 = sin(r17936);
        double r17938 = 1.0;
        double r17939 = a;
        double r17940 = cos(r17939);
        double r17941 = cos(r17936);
        double r17942 = r17940 * r17941;
        double r17943 = sin(r17939);
        double r17944 = r17943 * r17937;
        double r17945 = r17942 - r17944;
        double r17946 = r17938 / r17945;
        double r17947 = r17937 * r17946;
        double r17948 = r17935 * r17947;
        return r17948;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.3

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
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 div-inv 0.4

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

6. Final simplification 0.4

   \[\leadsto r \cdot \left(\sin b \cdot \frac{1}{\cos a \cdot \cos b - \sin a \cdot \sin b}\right)\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))