Average Error: 14.9 → 0.4
Time: 7.1s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\sin a} \cdot \left(\sqrt[3]{\sin b} \cdot {\left({\left(\sin b\right)}^{2} \cdot {\left(\sin a\right)}^{2}\right)}^{\frac{1}{3}}\right)}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\sin a} \cdot \left(\sqrt[3]{\sin b} \cdot {\left({\left(\sin b\right)}^{2} \cdot {\left(\sin a\right)}^{2}\right)}^{\frac{1}{3}}\right)}
double f(double r, double a, double b) {
        double r16538 = r;
        double r16539 = b;
        double r16540 = sin(r16539);
        double r16541 = a;
        double r16542 = r16541 + r16539;
        double r16543 = cos(r16542);
        double r16544 = r16540 / r16543;
        double r16545 = r16538 * r16544;
        return r16545;
}
↓
double f(double r, double a, double b) {
        double r16546 = r;
        double r16547 = b;
        double r16548 = sin(r16547);
        double r16549 = a;
        double r16550 = cos(r16549);
        double r16551 = cos(r16547);
        double r16552 = r16550 * r16551;
        double r16553 = sin(r16549);
        double r16554 = cbrt(r16553);
        double r16555 = cbrt(r16548);
        double r16556 = 2.0;
        double r16557 = pow(r16548, r16556);
        double r16558 = pow(r16553, r16556);
        double r16559 = r16557 * r16558;
        double r16560 = 0.3333333333333333;
        double r16561 = pow(r16559, r16560);
        double r16562 = r16555 * r16561;
        double r16563 = r16554 * r16562;
        double r16564 = r16552 - r16563;
        double r16565 = r16548 / r16564;
        double r16566 = r16546 * r16565;
        return r16566;
}

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 14.9

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

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

  10. Applied cube-mult0.4

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

  11. Applied cbrt-prod0.4

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

  13. Applied cbrt-prod0.4

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

  14. Applied associate-*l*0.4

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

  15. Simplified0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\sin a} \cdot \color{blue}{\left(\sqrt[3]{\sin b} \cdot {\left({\left(\sin b\right)}^{2} \cdot {\left(\sin a\right)}^{2}\right)}^{\frac{1}{3}}\right)}}\]

  16. Final simplification0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\sin a} \cdot \left(\sqrt[3]{\sin b} \cdot {\left({\left(\sin b\right)}^{2} \cdot {\left(\sin a\right)}^{2}\right)}^{\frac{1}{3}}\right)}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))