Average Error: 0.1 → 0.2
Time: 4.8s
Precision: binary64
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(e, \cos v, 1\right)}\\ \frac{\frac{e \cdot \sin v}{t_0}}{t_0} \end{array} \]
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (let* ((t_0 (sqrt (fma e (cos v) 1.0)))) (/ (/ (* e (sin v)) t_0) t_0)))
double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}
↓
double code(double e, double v) {
	double t_0 = sqrt(fma(e, cos(v), 1.0));
	return ((e * sin(v)) / t_0) / t_0;
}

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end
↓
function code(e, v)
	t_0 = sqrt(fma(e, cos(v), 1.0))
	return Float64(Float64(Float64(e * sin(v)) / t_0) / t_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_] := Block[{t$95$0 = N[Sqrt[N[(e * N[Cos[v], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\begin{array}{l}
t_0 := \sqrt{\mathsf{fma}\left(e, \cos v, 1\right)}\\
\frac{\frac{e \cdot \sin v}{t_0}}{t_0}
\end{array}

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 add-sqr-sqrt_binary640.2

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

  4. Applied associate-/r*_binary640.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022137 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))