Trigonometry A

Average Error: 0.1 → 0.1

Time: 18.7s

Precision: 64

\[0.0 \le e \le 1\]

Math

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

↓

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

C

double f(double e, double v) {
        double r22162 = e;
        double r22163 = v;
        double r22164 = sin(r22163);
        double r22165 = r22162 * r22164;
        double r22166 = 1.0;
        double r22167 = cos(r22163);
        double r22168 = r22162 * r22167;
        double r22169 = r22166 + r22168;
        double r22170 = r22165 / r22169;
        return r22170;
}

↓

double f(double e, double v) {
        double r22171 = e;
        double r22172 = v;
        double r22173 = sin(r22172);
        double r22174 = r22171 * r22173;
        double r22175 = 1.0;
        double r22176 = cos(r22172);
        double r22177 = r22171 * r22176;
        double r22178 = r22175 + r22177;
        double r22179 = r22174 / r22178;
        return r22179;
}

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 pow1 0.1

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

4. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))