Average Error: 14.7 → 0.4
Time: 26.0s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)
double f(double r, double a, double b) {
        double r25083 = r;
        double r25084 = b;
        double r25085 = sin(r25084);
        double r25086 = a;
        double r25087 = r25086 + r25084;
        double r25088 = cos(r25087);
        double r25089 = r25085 / r25088;
        double r25090 = r25083 * r25089;
        return r25090;
}
↓
double f(double r, double a, double b) {
        double r25091 = r;
        double r25092 = b;
        double r25093 = sin(r25092);
        double r25094 = a;
        double r25095 = cos(r25094);
        double r25096 = cos(r25092);
        double r25097 = r25095 * r25096;
        double r25098 = 3.0;
        double r25099 = pow(r25097, r25098);
        double r25100 = sin(r25094);
        double r25101 = r25100 * r25093;
        double r25102 = pow(r25101, r25098);
        double r25103 = r25099 - r25102;
        double r25104 = r25093 / r25103;
        double r25105 = r25091 * r25104;
        double r25106 = r25097 * r25097;
        double r25107 = r25101 * r25101;
        double r25108 = r25097 * r25101;
        double r25109 = r25107 + r25108;
        double r25110 = r25106 + r25109;
        double r25111 = r25105 * r25110;
        return r25111;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

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

  3. Applied cos-sum0.3

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

  5. Applied flip3--0.4

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}}\]

  6. Applied associate-/r/0.4

    \[\leadsto r \cdot \color{blue}{\left(\frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}} \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\right)}\]

  7. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)}\]

  8. Final simplification0.4

    \[\leadsto \left(r \cdot \frac{\sin b}{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}\right) \cdot \left(\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))