Average Error: 0.2 → 0.1
Time: 2.6s
Precision: binary64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
↓
\[\begin{array}{l} t_0 := \log \left(\sqrt{e^{\frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}}}\right)\\ \log \left({\left(e^{t_0 + t_0}\right)}^{6}\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (sqrt (exp (/ (+ x -1.0) (+ x (fma 4.0 (sqrt x) 1.0))))))))
   (log (pow (exp (+ t_0 t_0)) 6.0))))
double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}
↓
double code(double x) {
	double t_0 = log(sqrt(exp(((x + -1.0) / (x + fma(4.0, sqrt(x), 1.0))))));
	return log(pow(exp((t_0 + t_0)), 6.0));
}

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 = log(sqrt(exp(Float64(Float64(x + -1.0) / Float64(x + fma(4.0, sqrt(x), 1.0))))))
	return log((exp(Float64(t_0 + t_0)) ^ 6.0))
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[Log[N[Sqrt[N[Exp[N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[Log[N[Power[N[Exp[N[(t$95$0 + t$95$0), $MachinePrecision]], $MachinePrecision], 6.0], $MachinePrecision]], $MachinePrecision]]

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

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 0.2

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

  2. Applied egg-rr0.1

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022160 
(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)))))