Average Error: 0.1 → 0.6
Time: 4.4s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[\sin v \cdot \left(e + \cos v \cdot \left({e}^{3} - e \cdot e\right)\right) \]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\sin v \cdot \left(e + \cos v \cdot \left({e}^{3} - e \cdot e\right)\right)

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (* (sin v) (+ e (* (cos v) (- (pow e 3.0) (* e e))))))
double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}
↓
double code(double e, double v) {
	return sin(v) * (e + (cos(v) * (pow(e, 3.0) - (e * e))));
}

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. Simplified0.1

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

  3. Taylor expanded in e around 0 0.5

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

  4. Simplified0.5

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

  5. Taylor expanded in v around 0 0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2021313 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))