Average Error: 0.1 → 0.1
Time: 31.8s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
double f(double e, double v) {
        double r19839 = e;
        double r19840 = v;
        double r19841 = sin(r19840);
        double r19842 = r19839 * r19841;
        double r19843 = 1.0;
        double r19844 = cos(r19840);
        double r19845 = r19839 * r19844;
        double r19846 = r19843 + r19845;
        double r19847 = r19842 / r19846;
        return r19847;
}
↓
double f(double e, double v) {
        double r19848 = e;
        double r19849 = v;
        double r19850 = sin(r19849);
        double r19851 = r19848 * r19850;
        double r19852 = 1.0;
        double r19853 = cos(r19849);
        double r19854 = r19848 * r19853;
        double r19855 = r19852 + r19854;
        double r19856 = r19851 / r19855;
        return r19856;
}

Error

Bits error versus e

Bits error versus v

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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. Using strategy rm

  6. Applied add-sqr-sqrt0.4

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

  7. Applied sqrt-prod0.6

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

  8. Applied associate-*l*0.6

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

  9. Taylor expanded around inf 0.1

    \[\leadsto \frac{\color{blue}{e \cdot \sin v}}{1 + e \cdot \cos v}\]

  10. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))