Trigonometry A

Average Error: 0.1 → 0.2

Time: 54.0s

Precision: 64

\[0 \le e \le 1\]

Math

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

↓

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

C

double f(double e, double v) {
        double r3013455 = e;
        double r3013456 = v;
        double r3013457 = sin(r3013456);
        double r3013458 = r3013455 * r3013457;
        double r3013459 = 1.0;
        double r3013460 = cos(r3013456);
        double r3013461 = r3013455 * r3013460;
        double r3013462 = r3013459 + r3013461;
        double r3013463 = r3013458 / r3013462;
        return r3013463;
}

↓

double f(double e, double v) {
        double r3013464 = e;
        double r3013465 = v;
        double r3013466 = sin(r3013465);
        double r3013467 = r3013464 * r3013466;
        double r3013468 = cos(r3013465);
        double r3013469 = r3013468 * r3013464;
        double r3013470 = 1.0;
        double r3013471 = r3013469 + r3013470;
        double r3013472 = sqrt(r3013471);
        double r3013473 = r3013467 / r3013472;
        double r3013474 = r3013473 / r3013472;
        return r3013474;
}

Error

Bits error versus e

Bits error versus v

Derivation

1. Initial program 0.1

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

3. Applied add-sqr-sqrt 0.2

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

4. Applied associate-/r* 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019119 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))