Average Error: 0.1 → 0.1
Time: 5.3s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\sin v \cdot \frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\sin v \cdot \frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)}
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 / fma(cos(v), e, 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. Simplified0.2

    \[\leadsto \color{blue}{\frac{\sin v}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{e}}}\]
  3. Using strategy rm

  4. Applied div-inv0.2

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

  5. Simplified0.1

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

  6. Final simplification0.1

    \[\leadsto \sin v \cdot \frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))