Average Error: 0.1 → 0.1
Time: 6.7s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{1 + e \cdot \cos v} \cdot \sin v\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{1 + e \cdot \cos v} \cdot \sin v
double f(double e, double v) {
        double r11456 = e;
        double r11457 = v;
        double r11458 = sin(r11457);
        double r11459 = r11456 * r11458;
        double r11460 = 1.0;
        double r11461 = cos(r11457);
        double r11462 = r11456 * r11461;
        double r11463 = r11460 + r11462;
        double r11464 = r11459 / r11463;
        return r11464;
}
↓
double f(double e, double v) {
        double r11465 = e;
        double r11466 = 1.0;
        double r11467 = v;
        double r11468 = cos(r11467);
        double r11469 = r11465 * r11468;
        double r11470 = r11466 + r11469;
        double r11471 = r11465 / r11470;
        double r11472 = sin(r11467);
        double r11473 = r11471 * r11472;
        return r11473;
}

Error

Bits error versus e

Bits error versus v

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.1

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

  3. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.3

    \[\leadsto \frac{e}{\frac{1 + e \cdot \cos v}{\color{blue}{1 \cdot \sin v}}}\]

  6. Applied *-un-lft-identity0.3

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

  7. Applied times-frac0.3

    \[\leadsto \frac{e}{\color{blue}{\frac{1}{1} \cdot \frac{1 + e \cdot \cos v}{\sin v}}}\]

  8. Applied *-un-lft-identity0.3

    \[\leadsto \frac{\color{blue}{1 \cdot e}}{\frac{1}{1} \cdot \frac{1 + e \cdot \cos v}{\sin v}}\]

  9. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{1}{\frac{1}{1}} \cdot \frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}}\]

  10. Simplified0.3

    \[\leadsto \color{blue}{1} \cdot \frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\]

  11. Simplified0.1

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

  12. Final simplification0.1

    \[\leadsto \frac{e}{1 + e \cdot \cos v} \cdot \sin v\]

Reproduce

herbie shell --seed 2020045 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))