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

Average Error: 14.7 → 0.4

Time: 26.4s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r908541 = r;
        double r908542 = b;
        double r908543 = sin(r908542);
        double r908544 = r908541 * r908543;
        double r908545 = a;
        double r908546 = r908545 + r908542;
        double r908547 = cos(r908546);
        double r908548 = r908544 / r908547;
        return r908548;
}

↓

double f(double r, double a, double b) {
        double r908549 = 1.0;
        double r908550 = b;
        double r908551 = cos(r908550);
        double r908552 = a;
        double r908553 = cos(r908552);
        double r908554 = r908551 * r908553;
        double r908555 = sin(r908550);
        double r908556 = sin(r908552);
        double r908557 = r908555 * r908556;
        double r908558 = r908554 - r908557;
        double r908559 = r908549 / r908558;
        double r908560 = r;
        double r908561 = r908560 * r908555;
        double r908562 = r908559 * r908561;
        return r908562;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.7

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

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))