Trigonometry A

Average Error: 0.1 → 0.1

Time: 15.2s

Precision: 64

\[0.0 \le e \le 1\]

Math

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

↓

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

C

double f(double e, double v) {
        double r20073 = e;
        double r20074 = v;
        double r20075 = sin(r20074);
        double r20076 = r20073 * r20075;
        double r20077 = 1.0;
        double r20078 = cos(r20074);
        double r20079 = r20073 * r20078;
        double r20080 = r20077 + r20079;
        double r20081 = r20076 / r20080;
        return r20081;
}

↓

double f(double e, double v) {
        double r20082 = e;
        double r20083 = v;
        double r20084 = sin(r20083);
        double r20085 = r20082 * r20084;
        double r20086 = cos(r20083);
        double r20087 = r20086 * r20082;
        double r20088 = 1.0;
        double r20089 = r20087 + r20088;
        double r20090 = r20085 / r20089;
        return r20090;
}

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 +-commutative 0.1

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

4. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))