Average Error: 0.2 → 0.0
Time: 3.7s
Precision: binary64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(4, \sqrt{x}, x + 1\right)\\ \mathsf{fma}\left(6, \frac{x}{t_0}, \frac{-6}{t_0}\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 4.0 (sqrt x) (+ x 1.0)))) (fma 6.0 (/ x t_0) (/ -6.0 t_0))))
double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}
↓
double code(double x) {
	double t_0 = fma(4.0, sqrt(x), (x + 1.0));
	return fma(6.0, (x / t_0), (-6.0 / t_0));
}

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end
↓
function code(x)
	t_0 = fma(4.0, sqrt(x), Float64(x + 1.0))
	return fma(6.0, Float64(x / t_0), Float64(-6.0 / t_0))
end

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(6.0 * N[(x / t$95$0), $MachinePrecision] + N[(-6.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]

\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(4, \sqrt{x}, x + 1\right)\\
\mathsf{fma}\left(6, \frac{x}{t_0}, \frac{-6}{t_0}\right)
\end{array}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}} \]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

  3. Taylor expanded in x around 0 0.2

    \[\leadsto \frac{\color{blue}{6 \cdot x - 6}}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \]

  4. Applied egg-rr0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(6, \frac{x}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}, -\frac{6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\right)} \]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(6, \frac{x}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}, \frac{-6}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\right) \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))