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

Average Error: 15.1 → 0.4

Time: 6.7s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r17102 = r;
        double r17103 = b;
        double r17104 = sin(r17103);
        double r17105 = r17102 * r17104;
        double r17106 = a;
        double r17107 = r17106 + r17103;
        double r17108 = cos(r17107);
        double r17109 = r17105 / r17108;
        return r17109;
}

↓

double f(double r, double a, double b) {
        double r17110 = r;
        double r17111 = b;
        double r17112 = sin(r17111);
        double r17113 = r17110 * r17112;
        double r17114 = a;
        double r17115 = cos(r17114);
        double r17116 = cos(r17111);
        double r17117 = r17115 * r17116;
        double r17118 = sin(r17114);
        double r17119 = r17118 * r17112;
        double r17120 = exp(r17119);
        double r17121 = log(r17120);
        double r17122 = r17117 - r17121;
        double r17123 = r17113 / r17122;
        return r17123;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

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

3. Applied cos-sum 0.3

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

5. Applied add-log-exp 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020100 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))