Average Error: 0.1 → 0.1
Time: 4.9s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\sin v \cdot \frac{e}{{\left(e \cdot \cos v\right)}^{3} + {1}^{3}}\right) \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - \left(e \cdot \cos v\right) \cdot 1\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\sin v \cdot \frac{e}{{\left(e \cdot \cos v\right)}^{3} + {1}^{3}}\right) \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - \left(e \cdot \cos v\right) \cdot 1\right)\right)
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (*
  (* (sin v) (/ e (+ (pow (* e (cos v)) 3.0) (pow 1.0 3.0))))
  (+ (* 1.0 1.0) (- (* (* e (cos v)) (* e (cos v))) (* (* e (cos v)) 1.0)))))
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) (((double) sin(v)) * (e / ((double) (((double) pow(((double) (e * ((double) cos(v)))), 3.0)) + ((double) pow(1.0, 3.0))))))) * ((double) (((double) (1.0 * 1.0)) + ((double) (((double) (((double) (e * ((double) cos(v)))) * ((double) (e * ((double) cos(v)))))) - ((double) (((double) (e * ((double) 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 flip3-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020198 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))