Average Error: 0.1 → 0.4
Time: 9.1s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\sqrt{e} \cdot \left(\sqrt{e} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\sqrt{e} \cdot \left(\sqrt{e} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)
double f(double e, double v) {
        double r10372 = e;
        double r10373 = v;
        double r10374 = sin(r10373);
        double r10375 = r10372 * r10374;
        double r10376 = 1.0;
        double r10377 = cos(r10373);
        double r10378 = r10372 * r10377;
        double r10379 = r10376 + r10378;
        double r10380 = r10375 / r10379;
        return r10380;
}
↓
double f(double e, double v) {
        double r10381 = e;
        double r10382 = sqrt(r10381);
        double r10383 = v;
        double r10384 = sin(r10383);
        double r10385 = cos(r10383);
        double r10386 = 1.0;
        double r10387 = fma(r10385, r10381, r10386);
        double r10388 = r10384 / r10387;
        double r10389 = r10382 * r10388;
        double r10390 = r10382 * r10389;
        return r10390;
}

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 add-sqr-sqrt0.4

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

  9. Applied associate-*l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))