Average Error: 0.1 → 0.1
Time: 4.8s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)}

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v) :precision binary64 (* e (/ (sin v) (fma e (cos v) 1.0))))
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) / fma(e, cos(v), 1.0));
}

Error

Bits error versus e

Bits error versus v

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]

  2. Simplified0.1

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

  3. Applied *-un-lft-identity_binary640.1

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

  4. Applied times-frac_binary640.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022020 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))