Average Error: 0.2 → 0.1
Time: 2.2s
Precision: binary64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
↓
\[\log \left({\left(e^{6}\right)}^{\left({\left(\frac{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}{x + -1}\right)}^{-1}\right)}\right) \]
(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))
↓
(FPCore (x)
 :precision binary64
 (log (pow (exp 6.0) (pow (/ (+ x (fma 4.0 (sqrt x) 1.0)) (+ x -1.0)) -1.0))))
double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}
↓
double code(double x) {
	return log(pow(exp(6.0), pow(((x + fma(4.0, sqrt(x), 1.0)) / (x + -1.0)), -1.0)));
}

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

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

Error

Bits error versus x

Target

Original0.2
Target0.1
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

    \[\leadsto 6 \cdot \frac{1}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)}{x + -1}\right)\right)}} \]

  4. Applied egg-rr0.0

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

  5. Applied egg-rr0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022166 
(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)))))