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

Average Error: 14.6 → 0.4

Time: 6.0s

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 r16327 = r;
        double r16328 = b;
        double r16329 = sin(r16328);
        double r16330 = r16327 * r16329;
        double r16331 = a;
        double r16332 = r16331 + r16328;
        double r16333 = cos(r16332);
        double r16334 = r16330 / r16333;
        return r16334;
}

↓

double f(double r, double a, double b) {
        double r16335 = r;
        double r16336 = b;
        double r16337 = sin(r16336);
        double r16338 = r16335 * r16337;
        double r16339 = a;
        double r16340 = cos(r16339);
        double r16341 = cos(r16336);
        double r16342 = r16340 * r16341;
        double r16343 = sin(r16339);
        double r16344 = r16343 * r16337;
        double r16345 = exp(r16344);
        double r16346 = log(r16345);
        double r16347 = r16342 - r16346;
        double r16348 = r16338 / r16347;
        return r16348;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.6

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
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 2020059 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))