Average Error: 15.2 → 0.4
Time: 6.7s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}}
double f(double r, double a, double b) {
        double r17184 = r;
        double r17185 = b;
        double r17186 = sin(r17185);
        double r17187 = a;
        double r17188 = r17187 + r17185;
        double r17189 = cos(r17188);
        double r17190 = r17186 / r17189;
        double r17191 = r17184 * r17190;
        return r17191;
}
↓
double f(double r, double a, double b) {
        double r17192 = r;
        double r17193 = b;
        double r17194 = sin(r17193);
        double r17195 = a;
        double r17196 = cos(r17195);
        double r17197 = cos(r17193);
        double r17198 = r17196 * r17197;
        double r17199 = sin(r17195);
        double r17200 = r17199 * r17194;
        double r17201 = 3.0;
        double r17202 = pow(r17200, r17201);
        double r17203 = cbrt(r17202);
        double r17204 = r17198 - r17203;
        double r17205 = r17194 / r17204;
        double r17206 = r17192 * r17205;
        return r17206;
}

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 15.2

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

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

Reproduce

herbie shell --seed 2020027 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))