Average Error: 0.1 → 0.3
Time: 9.3s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{e} \cdot \left(1 - e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\sin v}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{e} \cdot \left(1 - e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)
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) (((double) sin(v)) / ((double) (((double) (((double) fma(((double) cos(v)), e, 1.0)) / e)) * ((double) (1.0 - ((double) (e * ((double) cos(v)))))))))) * ((double) (1.0 - ((double) (e * ((double) 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 flip-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Simplified0.1

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

  7. Applied clear-num0.2

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

  8. Applied frac-times0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))