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

Average Error: 15.1 → 0.4

Time: 6.7s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r16805 = r;
        double r16806 = b;
        double r16807 = sin(r16806);
        double r16808 = a;
        double r16809 = r16808 + r16806;
        double r16810 = cos(r16809);
        double r16811 = r16807 / r16810;
        double r16812 = r16805 * r16811;
        return r16812;
}

↓

double f(double r, double a, double b) {
        double r16813 = r;
        double r16814 = b;
        double r16815 = sin(r16814);
        double r16816 = a;
        double r16817 = cos(r16816);
        double r16818 = cos(r16814);
        double r16819 = r16817 * r16818;
        double r16820 = sin(r16816);
        double r16821 = r16820 * r16815;
        double r16822 = exp(r16821);
        double r16823 = log(r16822);
        double r16824 = r16819 - r16823;
        double r16825 = r16815 / r16824;
        double r16826 = r16813 * r16825;
        return r16826;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 15.1

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
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 add-log-exp 0.4

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020035 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))