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

Average Error: 14.9 → 0.4

Time: 29.6s

Precision: 64

Math

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

↓

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

C

double f(double r, double a, double b) {
        double r582117 = r;
        double r582118 = b;
        double r582119 = sin(r582118);
        double r582120 = a;
        double r582121 = r582120 + r582118;
        double r582122 = cos(r582121);
        double r582123 = r582119 / r582122;
        double r582124 = r582117 * r582123;
        return r582124;
}

↓

double f(double r, double a, double b) {
        double r582125 = 1.0;
        double r582126 = a;
        double r582127 = cos(r582126);
        double r582128 = b;
        double r582129 = cos(r582128);
        double r582130 = r582127 * r582129;
        double r582131 = sin(r582128);
        double r582132 = sin(r582126);
        double r582133 = r582131 * r582132;
        double r582134 = r582130 - r582133;
        double r582135 = r582125 / r582134;
        double r582136 = r582135 * r582131;
        double r582137 = r;
        double r582138 = r582136 * r582137;
        return r582138;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.9

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

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

6. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019152 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))