Average Error: 0.1 → 0.1
Time: 5.9s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + \sqrt[3]{\cos v} \cdot \left(e \cdot \left(\sqrt[3]{\cos v} \cdot \sqrt[3]{\cos v}\right)\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 + \sqrt[3]{\cos v} \cdot \left(e \cdot \left(\sqrt[3]{\cos v} \cdot \sqrt[3]{\cos v}\right)\right)}
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (/
  (* e (sin v))
  (+ 1.0 (* (cbrt (cos v)) (* e (* (cbrt (cos v)) (cbrt (cos v))))))))
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)) / (1.0 + (cbrt(cos(v)) * (e * (cbrt(cos(v)) * cbrt(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 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt_binary64_11360.1

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

  4. Applied associate-*r*_binary64_10410.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2021058 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))