Average Error: 0.1 → 0.1
Time: 5.0s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[\frac{e \cdot \sin v}{e \cdot \cos v + 1} \]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{e \cdot \cos v + 1}

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ (* 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 (e * sin(v)) / ((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 div-inv_binary640.1

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

  4. Simplified0.1

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

  5. Taylor expanded around inf 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2021198 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))