Average Error: 0.1 → 0.1
Time: 19.7s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r1012963 = e;
        double r1012964 = v;
        double r1012965 = sin(r1012964);
        double r1012966 = r1012963 * r1012965;
        double r1012967 = 1.0;
        double r1012968 = cos(r1012964);
        double r1012969 = r1012963 * r1012968;
        double r1012970 = r1012967 + r1012969;
        double r1012971 = r1012966 / r1012970;
        return r1012971;
}
↓
double f(double e, double v) {
        double r1012972 = e;
        double r1012973 = v;
        double r1012974 = sin(r1012973);
        double r1012975 = cos(r1012973);
        double r1012976 = 1.0;
        double r1012977 = fma(r1012975, r1012972, r1012976);
        double r1012978 = r1012974 / r1012977;
        double r1012979 = r1012972 * r1012978;
        return r1012979;
}

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

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied *-un-lft-identity0.1

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))