Average Error: 0.1 → 0.1
Time: 17.4s
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 r18092 = e;
        double r18093 = v;
        double r18094 = sin(r18093);
        double r18095 = r18092 * r18094;
        double r18096 = 1.0;
        double r18097 = cos(r18093);
        double r18098 = r18092 * r18097;
        double r18099 = r18096 + r18098;
        double r18100 = r18095 / r18099;
        return r18100;
}
↓
double f(double e, double v) {
        double r18101 = e;
        double r18102 = v;
        double r18103 = sin(r18102);
        double r18104 = -r18103;
        double r18105 = cos(r18102);
        double r18106 = 1.0;
        double r18107 = fma(r18105, r18101, r18106);
        double r18108 = -r18107;
        double r18109 = r18104 / r18108;
        double r18110 = r18101 * r18109;
        return r18110;
}

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. Using strategy rm

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

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

  5. Applied times-frac0.1

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

  6. Simplified0.1

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

  8. Applied frac-2neg0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))