Average Error: 58.6 → 0.3
Time: 5.9s
Precision: binary64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
↓
\[0.5 \cdot \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) \]
(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))
↓
(FPCore (x)
 :precision binary64
 (* 0.5 (fma 2.0 x (* 0.6666666666666666 (pow x 3.0)))))
double code(double x) {
	return (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
}
↓
double code(double x) {
	return 0.5 * fma(2.0, x, (0.6666666666666666 * pow(x, 3.0)));
}

function code(x)
	return Float64(Float64(1.0 / 2.0) * log(Float64(Float64(1.0 + x) / Float64(1.0 - x))))
end
↓
function code(x)
	return Float64(0.5 * fma(2.0, x, Float64(0.6666666666666666 * (x ^ 3.0))))
end

code[x_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[Log[N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(0.5 * N[(2.0 * x + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
0.5 \cdot \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)

Error

Bits error versus x

Derivation

  1. Initial program 58.6

    \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]

  2. Simplified0.0

    \[\leadsto \color{blue}{0.5 \cdot \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right)} \]

  3. Taylor expanded in x around 0 0.3

    \[\leadsto 0.5 \cdot \color{blue}{\left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]

  4. Simplified0.3

    \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)} \]

  5. Final simplification0.3

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))