Average Error: 0.1 → 0.1
Time: 1.2min
Precision: binary64
Cost: 52736
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\sqrt{1 - e \cdot \cos v} \cdot \frac{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \sin v}{\sqrt{1 - e \cdot \left(e \cdot {\cos v}^{2}\right)}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\sqrt{1 - e \cdot \cos v} \cdot \frac{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \sin v}{\sqrt{1 - e \cdot \left(e \cdot {\cos v}^{2}\right)}}
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (*
  (sqrt (- 1.0 (* e (cos v))))
  (/
   (* (/ e (sqrt (+ 1.0 (* e (cos v))))) (sin v))
   (sqrt (- 1.0 (* e (* e (pow (cos v) 2.0))))))))
double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}
↓
double code(double e, double v) {
	return sqrt(1.0 - (e * cos(v))) * (((e / sqrt(1.0 + (e * cos(v)))) * sin(v)) / sqrt(1.0 - (e * (e * pow(cos(v), 2.0)))));
}

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.1
Cost13376
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
Alternative 2
Error0.8
Cost6848
\[\frac{e \cdot \sin v}{1 + e}\]
Alternative 3
Error1.4
Cost6592
\[e \cdot \sin v\]
Alternative 4
Error31.1
Cost576
\[\left(e \cdot v\right) \cdot \frac{1}{1 + e}\]
Alternative 5
Error31.1
Cost448
\[\frac{e \cdot v}{1 + e}\]
Alternative 6
Error31.1
Cost448
\[\frac{e}{\frac{1 + e}{v}}\]
Alternative 7
Error31.3
Cost448
\[e \cdot \left(v - e \cdot v\right)\]
Alternative 8
Error31.6
Cost192
\[e \cdot v\]
Alternative 9
Error45.9
Cost64
\[0\]
Alternative 10
Error61.1
Cost64
\[v\]

Error

Derivation

  1. Initial program 0.1

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

  3. Applied add-sqr-sqrt_binary64_14640.2

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

  4. Applied associate-/r*_binary64_13860.2

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

  5. Simplified0.2

    \[\leadsto \frac{\color{blue}{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \sin v}}{\sqrt{1 + e \cdot \cos v}}\]
  6. Using strategy rm

  7. Applied flip-+_binary64_14160.1

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

  8. Applied sqrt-div_binary64_14590.1

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

  9. Applied associate-/r/_binary64_13880.1

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

  10. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \sin v}{\sqrt{1 - e \cdot \left(e \cdot {\cos v}^{2}\right)}}} \cdot \sqrt{1 - e \cdot \cos v}\]

  11. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{1 - e \cdot \cos v} \cdot \frac{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \sin v}{\sqrt{1 - e \cdot \left(e \cdot {\cos v}^{2}\right)}}}\]

  12. Final simplification0.1

    \[\leadsto \sqrt{1 - e \cdot \cos v} \cdot \frac{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \sin v}{\sqrt{1 - e \cdot \left(e \cdot {\cos v}^{2}\right)}}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))