Average Error: 0.1 → 0.1
Time: 4.1s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[\sin v \cdot \frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)} \]
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v) :precision binary64 (* (sin v) (/ e (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 sin(v) * (e / fma(e, cos(v), 1.0));
}

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end
↓
function code(e, v)
	return Float64(sin(v) * Float64(e / fma(e, cos(v), 1.0)))
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[(N[Sin[v], $MachinePrecision] * N[(e / N[(e * N[Cos[v], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2022160 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))