Average Error: 0.1 → 0.1
Time: 5.2s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)
double f(double e, double v) {
        double r12307 = e;
        double r12308 = v;
        double r12309 = sin(r12308);
        double r12310 = r12307 * r12309;
        double r12311 = 1.0;
        double r12312 = cos(r12308);
        double r12313 = r12307 * r12312;
        double r12314 = r12311 + r12313;
        double r12315 = r12310 / r12314;
        return r12315;
}
↓
double f(double e, double v) {
        double r12316 = e;
        double r12317 = v;
        double r12318 = sin(r12317);
        double r12319 = r12316 * r12318;
        double r12320 = 1.0;
        double r12321 = 3.0;
        double r12322 = pow(r12320, r12321);
        double r12323 = cos(r12317);
        double r12324 = r12316 * r12323;
        double r12325 = pow(r12324, r12321);
        double r12326 = r12322 + r12325;
        double r12327 = r12319 / r12326;
        double r12328 = r12320 * r12320;
        double r12329 = r12324 * r12324;
        double r12330 = r12320 * r12324;
        double r12331 = r12329 - r12330;
        double r12332 = r12328 + r12331;
        double r12333 = r12327 * r12332;
        return r12333;
}

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
  2. Using strategy rm

  3. Applied flip3-+0.1

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}}\]

  4. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)}\]

  5. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))