Average Error: 15.0 → 0.4
Time: 40.9s
Precision: 64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\sin a \cdot \sin b\right)}} \cdot r\]
\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)\right) \cdot \left(\sin a \cdot \sin b\right)}} \cdot r
double f(double r, double a, double b) {
        double r1447660 = r;
        double r1447661 = b;
        double r1447662 = sin(r1447661);
        double r1447663 = r1447660 * r1447662;
        double r1447664 = a;
        double r1447665 = r1447664 + r1447661;
        double r1447666 = cos(r1447665);
        double r1447667 = r1447663 / r1447666;
        return r1447667;
}
↓
double f(double r, double a, double b) {
        double r1447668 = b;
        double r1447669 = sin(r1447668);
        double r1447670 = a;
        double r1447671 = cos(r1447670);
        double r1447672 = cos(r1447668);
        double r1447673 = r1447671 * r1447672;
        double r1447674 = sin(r1447670);
        double r1447675 = r1447674 * r1447669;
        double r1447676 = r1447675 * r1447675;
        double r1447677 = r1447676 * r1447675;
        double r1447678 = cbrt(r1447677);
        double r1447679 = r1447673 - r1447678;
        double r1447680 = r1447669 / r1447679;
        double r1447681 = r;
        double r1447682 = r1447680 * r1447681;
        return r1447682;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.0

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

  3. Applied cos-sum0.3

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

  5. Applied *-un-lft-identity0.3

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  9. Applied add-cbrt-cube0.4

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

  10. Applied add-cbrt-cube0.4

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

  11. Applied cbrt-unprod0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))