Average Error: 14.7 → 0.3
Time: 24.6s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin a \cdot \sin b}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\cos b \cdot \cos a - \sin a \cdot \sin b}
double f(double r, double a, double b) {
        double r25702 = r;
        double r25703 = b;
        double r25704 = sin(r25703);
        double r25705 = a;
        double r25706 = r25705 + r25703;
        double r25707 = cos(r25706);
        double r25708 = r25704 / r25707;
        double r25709 = r25702 * r25708;
        return r25709;
}
↓
double f(double r, double a, double b) {
        double r25710 = r;
        double r25711 = b;
        double r25712 = sin(r25711);
        double r25713 = r25710 * r25712;
        double r25714 = cos(r25711);
        double r25715 = a;
        double r25716 = cos(r25715);
        double r25717 = r25714 * r25716;
        double r25718 = sin(r25715);
        double r25719 = r25718 * r25712;
        double r25720 = r25717 - r25719;
        double r25721 = r25713 / r25720;
        return r25721;
}

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 add-cbrt-cube0.4

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

  6. Applied add-cbrt-cube0.4

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

  7. Applied cbrt-unprod0.4

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

  8. Simplified0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\color{blue}{{\left(\sin b \cdot \sin a\right)}^{3}}}}\]
  9. Using strategy rm

  10. Applied pow10.4

    \[\leadsto r \cdot \color{blue}{{\left(\frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}\right)}^{1}}\]

  11. Applied pow10.4

    \[\leadsto \color{blue}{{r}^{1}} \cdot {\left(\frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}\right)}^{1}\]

  12. Applied pow-prod-down0.4

    \[\leadsto \color{blue}{{\left(r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}\right)}^{1}}\]

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))