Average Error: 15.2 → 0.4
Time: 7.5s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\left(\sin b \cdot r\right) \cdot \frac{\sqrt[3]{1}}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\left(\sin b \cdot r\right) \cdot \frac{\sqrt[3]{1}}{\cos a \cdot \cos b - \sin a \cdot \sin b}
double code(double r, double a, double b) {
	return ((double) (r * ((double) (((double) sin(b)) / ((double) cos(((double) (a + b))))))));
}
↓
double code(double r, double a, double b) {
	return ((double) (((double) (((double) sin(b)) * r)) * ((double) (((double) cbrt(1.0)) / ((double) (((double) (((double) cos(a)) * ((double) cos(b)))) - ((double) (((double) sin(a)) * ((double) sin(b))))))))));
}

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 15.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 div-inv0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied add-cube-cbrt0.4

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

  9. Applied times-frac0.4

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

  10. Applied associate-*r*0.4

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

  11. Applied associate-*r*0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))