Average Error: 0.1 → 0.1
Time: 4.7s
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 r11112 = e;
        double r11113 = v;
        double r11114 = sin(r11113);
        double r11115 = r11112 * r11114;
        double r11116 = 1.0;
        double r11117 = cos(r11113);
        double r11118 = r11112 * r11117;
        double r11119 = r11116 + r11118;
        double r11120 = r11115 / r11119;
        return r11120;
}
↓
double f(double e, double v) {
        double r11121 = e;
        double r11122 = v;
        double r11123 = sin(r11122);
        double r11124 = r11121 * r11123;
        double r11125 = 1.0;
        double r11126 = 3.0;
        double r11127 = pow(r11125, r11126);
        double r11128 = cos(r11122);
        double r11129 = r11121 * r11128;
        double r11130 = pow(r11129, r11126);
        double r11131 = r11127 + r11130;
        double r11132 = r11124 / r11131;
        double r11133 = r11125 * r11125;
        double r11134 = r11129 * r11129;
        double r11135 = r11125 * r11129;
        double r11136 = r11134 - r11135;
        double r11137 = r11133 + r11136;
        double r11138 = r11132 * r11137;
        return r11138;
}

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 2020056 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))