Average Error: 0.1 → 0.2
Time: 4.6s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{e}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\sin v}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{e}}
double code(double e, double v) {
	return ((double) (((double) (e * ((double) sin(v)))) / ((double) (1.0 + ((double) (e * ((double) cos(v))))))));
}
↓
double code(double e, double v) {
	return ((double) (((double) sin(v)) / ((double) (((double) fma(((double) cos(v)), e, 1.0)) / e))));
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))