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

Average Error: 15.5 → 0.4

Time: 44.9s

Precision: 64

Math

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

↓

\[\frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}\]

C

double f(double r, double a, double b) {
        double r912558 = r;
        double r912559 = b;
        double r912560 = sin(r912559);
        double r912561 = a;
        double r912562 = r912561 + r912559;
        double r912563 = cos(r912562);
        double r912564 = r912560 / r912563;
        double r912565 = r912558 * r912564;
        return r912565;
}

↓

double f(double r, double a, double b) {
        double r912566 = r;
        double r912567 = b;
        double r912568 = cos(r912567);
        double r912569 = a;
        double r912570 = cos(r912569);
        double r912571 = r912568 * r912570;
        double r912572 = sin(r912567);
        double r912573 = r912571 / r912572;
        double r912574 = sin(r912569);
        double r912575 = r912573 - r912574;
        double r912576 = r912566 / r912575;
        return r912576;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.5

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

3. Applied cos-sum 0.3

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

5. Applied *-un-lft-identity 0.3

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

6. Applied associate-*l* 0.3

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

7. Simplified 0.4

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

8. Final simplification 0.4

   \[\leadsto \frac{r}{\frac{\cos b \cdot \cos a}{\sin b} - \sin a}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))