Average Error: 0.1 → 0.1
Time: 4.8s
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 r11097 = e;
        double r11098 = v;
        double r11099 = sin(r11098);
        double r11100 = r11097 * r11099;
        double r11101 = 1.0;
        double r11102 = cos(r11098);
        double r11103 = r11097 * r11102;
        double r11104 = r11101 + r11103;
        double r11105 = r11100 / r11104;
        return r11105;
}
↓
double f(double e, double v) {
        double r11106 = e;
        double r11107 = v;
        double r11108 = sin(r11107);
        double r11109 = r11106 * r11108;
        double r11110 = 1.0;
        double r11111 = 3.0;
        double r11112 = pow(r11110, r11111);
        double r11113 = cos(r11107);
        double r11114 = r11106 * r11113;
        double r11115 = pow(r11114, r11111);
        double r11116 = r11112 + r11115;
        double r11117 = r11109 / r11116;
        double r11118 = r11110 * r11110;
        double r11119 = r11114 * r11114;
        double r11120 = r11110 * r11114;
        double r11121 = r11119 - r11120;
        double r11122 = r11118 + r11121;
        double r11123 = r11117 * r11122;
        return r11123;
}

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)))))