Average Error: 15.0 → 0.4
Time: 14.4s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}
double f(double r, double a, double b) {
        double r17313 = r;
        double r17314 = b;
        double r17315 = sin(r17314);
        double r17316 = a;
        double r17317 = r17316 + r17314;
        double r17318 = cos(r17317);
        double r17319 = r17315 / r17318;
        double r17320 = r17313 * r17319;
        return r17320;
}
↓
double f(double r, double a, double b) {
        double r17321 = r;
        double r17322 = b;
        double r17323 = sin(r17322);
        double r17324 = a;
        double r17325 = cos(r17324);
        double r17326 = cos(r17322);
        double r17327 = r17325 * r17326;
        double r17328 = sin(r17324);
        double r17329 = r17323 * r17328;
        double r17330 = 3.0;
        double r17331 = pow(r17329, r17330);
        double r17332 = cbrt(r17331);
        double r17333 = r17327 - r17332;
        double r17334 = r17323 / r17333;
        double r17335 = r17321 * r17334;
        return r17335;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.0

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020043 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))