Average Error: 0.1 → 0.1
Time: 6.6s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{{\left(e \cdot \sin v\right)}^{1}}{1 + e \cdot \cos v}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{{\left(e \cdot \sin v\right)}^{1}}{1 + e \cdot \cos v}
double f(double e, double v) {
        double r20370 = e;
        double r20371 = v;
        double r20372 = sin(r20371);
        double r20373 = r20370 * r20372;
        double r20374 = 1.0;
        double r20375 = cos(r20371);
        double r20376 = r20370 * r20375;
        double r20377 = r20374 + r20376;
        double r20378 = r20373 / r20377;
        return r20378;
}
↓
double f(double e, double v) {
        double r20379 = e;
        double r20380 = v;
        double r20381 = sin(r20380);
        double r20382 = r20379 * r20381;
        double r20383 = 1.0;
        double r20384 = pow(r20382, r20383);
        double r20385 = 1.0;
        double r20386 = cos(r20380);
        double r20387 = r20379 * r20386;
        double r20388 = r20385 + r20387;
        double r20389 = r20384 / r20388;
        return r20389;
}

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.4

    \[\leadsto \frac{\sqrt{e} \cdot \color{blue}{\left(\sin v \cdot {e}^{\frac{1}{2}}\right)}}{1 + e \cdot \cos v}\]
  6. Using strategy rm

  7. Applied sqr-pow0.5

    \[\leadsto \frac{\sqrt{e} \cdot \left(\sin v \cdot \color{blue}{\left({e}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {e}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}{1 + e \cdot \cos v}\]

  8. Applied associate-*r*0.5

    \[\leadsto \frac{\sqrt{e} \cdot \color{blue}{\left(\left(\sin v \cdot {e}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot {e}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}}{1 + e \cdot \cos v}\]

  9. Simplified0.5

    \[\leadsto \frac{\sqrt{e} \cdot \left(\color{blue}{\left(\sin v \cdot {e}^{\frac{1}{4}}\right)} \cdot {e}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}{1 + e \cdot \cos v}\]
  10. Using strategy rm

  11. Applied pow10.5

    \[\leadsto \frac{\sqrt{e} \cdot \color{blue}{{\left(\left(\sin v \cdot {e}^{\frac{1}{4}}\right) \cdot {e}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{1}}}{1 + e \cdot \cos v}\]

  12. Applied pow10.5

    \[\leadsto \frac{\color{blue}{{\left(\sqrt{e}\right)}^{1}} \cdot {\left(\left(\sin v \cdot {e}^{\frac{1}{4}}\right) \cdot {e}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{1}}{1 + e \cdot \cos v}\]

  13. Applied pow-prod-down0.5

    \[\leadsto \frac{\color{blue}{{\left(\sqrt{e} \cdot \left(\left(\sin v \cdot {e}^{\frac{1}{4}}\right) \cdot {e}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)}^{1}}}{1 + e \cdot \cos v}\]

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019318 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))