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

Average Error: 15.1 → 0.4

Time: 6.2s

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 r15521 = r;
        double r15522 = b;
        double r15523 = sin(r15522);
        double r15524 = r15521 * r15523;
        double r15525 = a;
        double r15526 = r15525 + r15522;
        double r15527 = cos(r15526);
        double r15528 = r15524 / r15527;
        return r15528;
}

↓

double f(double r, double a, double b) {
        double r15529 = r;
        double r15530 = b;
        double r15531 = sin(r15530);
        double r15532 = r15529 * r15531;
        double r15533 = a;
        double r15534 = cos(r15533);
        double r15535 = cos(r15530);
        double r15536 = r15534 * r15535;
        double r15537 = sin(r15533);
        double r15538 = r15537 * r15531;
        double r15539 = exp(r15538);
        double r15540 = log(r15539);
        double r15541 = r15536 - r15540;
        double r15542 = r15532 / r15541;
        return r15542;
}

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 2020035 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))