Average Error: 0.1 → 0.1
Time: 13.8s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r20150 = e;
        double r20151 = v;
        double r20152 = sin(r20151);
        double r20153 = r20150 * r20152;
        double r20154 = 1.0;
        double r20155 = cos(r20151);
        double r20156 = r20150 * r20155;
        double r20157 = r20154 + r20156;
        double r20158 = r20153 / r20157;
        return r20158;
}
↓
double f(double e, double v) {
        double r20159 = e;
        double r20160 = v;
        double r20161 = sin(r20160);
        double r20162 = r20159 * r20161;
        double r20163 = cos(r20160);
        double r20164 = 1.0;
        double r20165 = fma(r20163, r20159, r20164);
        double r20166 = r20162 / r20165;
        return r20166;
}

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. Simplified0.1

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

  3. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))