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

Average Error: 14.5 → 0.4

Time: 25.0s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r1141301 = r;
        double r1141302 = b;
        double r1141303 = sin(r1141302);
        double r1141304 = r1141301 * r1141303;
        double r1141305 = a;
        double r1141306 = r1141305 + r1141302;
        double r1141307 = cos(r1141306);
        double r1141308 = r1141304 / r1141307;
        return r1141308;
}

↓

double f(double r, double a, double b) {
        double r1141309 = r;
        double r1141310 = b;
        double r1141311 = sin(r1141310);
        double r1141312 = r1141309 * r1141311;
        double r1141313 = a;
        double r1141314 = cos(r1141313);
        double r1141315 = cos(r1141310);
        double r1141316 = r1141314 * r1141315;
        double r1141317 = sin(r1141313);
        double r1141318 = r1141311 * r1141317;
        double r1141319 = exp(r1141318);
        double r1141320 = log(r1141319);
        double r1141321 = r1141316 - r1141320;
        double r1141322 = r1141312 / r1141321;
        return r1141322;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.5

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

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

Reproduce

herbie shell --seed 2019158 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))