Average Error: 0.1 → 0.1
Time: 5.4s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\left({1}^{3} + {\left(e \cdot \cos v\right)}^{3}\right) \cdot 1} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{\left({1}^{3} + {\left(e \cdot \cos v\right)}^{3}\right) \cdot 1} \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 r12142 = e;
        double r12143 = v;
        double r12144 = sin(r12143);
        double r12145 = r12142 * r12144;
        double r12146 = 1.0;
        double r12147 = cos(r12143);
        double r12148 = r12142 * r12147;
        double r12149 = r12146 + r12148;
        double r12150 = r12145 / r12149;
        return r12150;
}
↓
double f(double e, double v) {
        double r12151 = e;
        double r12152 = v;
        double r12153 = sin(r12152);
        double r12154 = r12151 * r12153;
        double r12155 = 1.0;
        double r12156 = 3.0;
        double r12157 = pow(r12155, r12156);
        double r12158 = cos(r12152);
        double r12159 = r12151 * r12158;
        double r12160 = pow(r12159, r12156);
        double r12161 = r12157 + r12160;
        double r12162 = 1.0;
        double r12163 = r12161 * r12162;
        double r12164 = r12154 / r12163;
        double r12165 = r12155 * r12155;
        double r12166 = r12159 * r12159;
        double r12167 = r12155 * r12159;
        double r12168 = r12166 - r12167;
        double r12169 = r12165 + r12168;
        double r12170 = r12164 * r12169;
        return r12170;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.4

    \[\leadsto \frac{\sqrt{e} \cdot \color{blue}{\left(\sin v \cdot {e}^{\frac{1}{2}}\right)}}{1 + e \cdot \cos v}\]
  6. Using strategy rm

  7. Applied flip3-+0.4

    \[\leadsto \frac{\sqrt{e} \cdot \left(\sin v \cdot {e}^{\frac{1}{2}}\right)}{\color{blue}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}}\]

  8. Applied associate-/r/0.4

    \[\leadsto \color{blue}{\frac{\sqrt{e} \cdot \left(\sin v \cdot {e}^{\frac{1}{2}}\right)}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)}\]

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019344 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))