Average Error: 0.1 → 0.1
Time: 10.6s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 + \left({e}^{2} \cdot {\cos v}^{2} - e \cdot \cos v\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 + \left({e}^{2} \cdot {\cos v}^{2} - e \cdot \cos v\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 (pow (* e (cos v)) 3.0)))
  (+ 1.0 (- (* (pow e 2.0) (pow (cos v) 2.0)) (* e (cos v))))))
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 + pow((e * cos(v)), 3.0))) * (1.0 + ((pow(e, 2.0) * pow(cos(v), 2.0)) - (e * 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 flip3-+_binary64_14450.1

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

  4. Applied associate-/r/_binary64_13880.1

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

  5. Simplified0.1

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

  6. Taylor expanded around 0 0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2021075 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))