Average Error: 0.1 → 0.1
Time: 5.5s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(e \cdot \frac{\sin v}{1 \cdot 1 - e \cdot \left(e \cdot {\left(\cos v\right)}^{2}\right)}\right) \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(e \cdot \frac{\sin v}{1 \cdot 1 - e \cdot \left(e \cdot {\left(\cos v\right)}^{2}\right)}\right) \cdot \left(1 - e \cdot \cos v\right)
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (*
  (* e (/ (sin v) (- (* 1.0 1.0) (* e (* e (pow (cos v) 2.0))))))
  (- 1.0 (* e (cos v)))))
double code(double e, double v) {
	return (((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) (((double) (1.0 * 1.0)) - ((double) (e * ((double) (e * ((double) pow(((double) cos(v)), 2.0))))))))))) * ((double) (1.0 - ((double) (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 Error: 0.1 bits

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

  2. SimplifiedError: 0.1 bits

    \[\leadsto \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}}\]
  3. Using strategy rm

  4. Applied flip-+Error: 0.1 bits

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

  5. Applied associate-/r/Error: 0.1 bits

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

  6. Applied associate-*r*Error: 0.1 bits

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

  7. SimplifiedError: 0.1 bits

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

  8. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))