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

Average Error: 15.0 → 0.4

Time: 14.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r16761 = r;
        double r16762 = b;
        double r16763 = sin(r16762);
        double r16764 = r16761 * r16763;
        double r16765 = a;
        double r16766 = r16765 + r16762;
        double r16767 = cos(r16766);
        double r16768 = r16764 / r16767;
        return r16768;
}

↓

double f(double r, double a, double b) {
        double r16769 = r;
        double r16770 = b;
        double r16771 = sin(r16770);
        double r16772 = r16769 * r16771;
        double r16773 = 1.0;
        double r16774 = a;
        double r16775 = cos(r16774);
        double r16776 = cos(r16770);
        double r16777 = r16775 * r16776;
        double r16778 = sin(r16774);
        double r16779 = r16778 * r16771;
        double r16780 = r16777 - r16779;
        double r16781 = r16773 / r16780;
        double r16782 = r16772 * r16781;
        return r16782;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.0

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

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

Reproduce

herbie shell --seed 2020047 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))