Average Error: 0.1 → 0.2
Time: 5.2s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v \cdot \frac{e}{\sqrt{e \cdot \cos v + 1}}}{\sqrt{e \cdot \cos v + 1}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\sin v \cdot \frac{e}{\sqrt{e \cdot \cos v + 1}}}{\sqrt{e \cdot \cos v + 1}}
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (/
  (* (sin v) (/ e (sqrt (+ (* e (cos v)) 1.0))))
  (sqrt (+ (* 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 (sin(v) * (e / sqrt((e * cos(v)) + 1.0))) / sqrt((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 add-sqr-sqrt_binary640.2

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\sqrt{1 + e \cdot \cos v} \cdot \sqrt{1 + e \cdot \cos v}}}\]

  4. Applied associate-/r*_binary640.2

    \[\leadsto \color{blue}{\frac{\frac{e \cdot \sin v}{\sqrt{1 + e \cdot \cos v}}}{\sqrt{1 + e \cdot \cos v}}}\]

  5. Simplified0.2

    \[\leadsto \frac{\color{blue}{\sin v \cdot \frac{e}{\sqrt{e \cdot \cos v + 1}}}}{\sqrt{1 + e \cdot \cos v}}\]

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020281 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))