Average Error: 0.1 → 0.1
Time: 4.8s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[e \cdot \left(\frac{1}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \sin v\right) \]
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (* e (* (/ 1.0 (fma e (cos v) 1.0)) (sin v))))
double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}
↓
double code(double e, double v) {
	return e * ((1.0 / fma(e, cos(v), 1.0)) * sin(v));
}

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end
↓
function code(e, v)
	return Float64(e * Float64(Float64(1.0 / fma(e, cos(v), 1.0)) * sin(v)))
end

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[e_, v_] := N[(e * N[(N[(1.0 / N[(e * N[Cos[v], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \left(\frac{1}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \sin v\right)

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 associate-/l*_binary640.3

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

  4. Applied associate-/r/_binary640.1

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

  5. Applied div-inv_binary640.1

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

  6. Applied associate-*l*_binary640.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022129 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))