Average Error: 0.1 → 0.1
Time: 4.8s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(e \cdot \sin v\right) \cdot \left(\frac{1}{\sqrt[3]{{\left(e \cdot \cos v\right)}^{6}} + -1} \cdot \left(e \cdot \cos v + -1\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(e \cdot \sin v\right) \cdot \left(\frac{1}{\sqrt[3]{{\left(e \cdot \cos v\right)}^{6}} + -1} \cdot \left(e \cdot \cos v + -1\right)\right)
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (*
  (* e (sin v))
  (* (/ 1.0 (+ (cbrt (pow (* e (cos v)) 6.0)) -1.0)) (+ (* e (cos v)) -1.0))))
double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}
↓
double code(double e, double v) {
	return (e * sin(v)) * ((1.0 / (cbrt(pow((e * cos(v)), 6.0)) + -1.0)) * ((e * cos(v)) + -1.0));
}

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 div-inv_binary640.1

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

  4. Simplified0.1

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

  6. Applied flip-+_binary640.1

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

  7. Applied associate-/r/_binary640.1

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

  8. Simplified0.1

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

  10. Applied add-cbrt-cube_binary640.1

    \[\leadsto \left(e \cdot \sin v\right) \cdot \left(\frac{1}{\color{blue}{\sqrt[3]{\left(\left(\left(e \cdot e\right) \cdot {\cos v}^{2}\right) \cdot \left(\left(e \cdot e\right) \cdot {\cos v}^{2}\right)\right) \cdot \left(\left(e \cdot e\right) \cdot {\cos v}^{2}\right)}} + -1} \cdot \left(e \cdot \cos v - 1\right)\right)\]

  11. Simplified0.1

    \[\leadsto \left(e \cdot \sin v\right) \cdot \left(\frac{1}{\sqrt[3]{\color{blue}{{\left(\cos v \cdot e\right)}^{6}}} + -1} \cdot \left(e \cdot \cos v - 1\right)\right)\]

  12. Final simplification0.1

    \[\leadsto \left(e \cdot \sin v\right) \cdot \left(\frac{1}{\sqrt[3]{{\left(e \cdot \cos v\right)}^{6}} + -1} \cdot \left(e \cdot \cos v + -1\right)\right)\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))