Average Error: 0.1 → 0.1
Time: 5.5s
Precision: binary64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + \left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right) \cdot \left(\sqrt[3]{e} \cdot \cos v\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 + \left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right) \cdot \left(\sqrt[3]{e} \cdot \cos v\right)}
double code(double e, double v) {
	return ((double) (((double) (e * ((double) sin(v)))) / ((double) (1.0 + ((double) (e * ((double) cos(v))))))));
}
↓
double code(double e, double v) {
	return ((double) (((double) (e * ((double) sin(v)))) / ((double) (1.0 + ((double) (((double) (((double) cbrt(e)) * ((double) cbrt(e)))) * ((double) (((double) cbrt(e)) * ((double) cos(v))))))))));
}

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-cube-cbrt0.1

    \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\left(\left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right) \cdot \sqrt[3]{e}\right)} \cdot \cos v}\]

  4. Applied associate-*l*0.1

    \[\leadsto \frac{e \cdot \sin v}{1 + \color{blue}{\left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right) \cdot \left(\sqrt[3]{e} \cdot \cos v\right)}}\]

  5. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{1 + \left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right) \cdot \left(\sqrt[3]{e} \cdot \cos v\right)}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))