Average Error: 0.2 → 0.2
Time: 3.1s
Precision: binary64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
↓
\[\frac{\mathsf{fma}\left(-6, x, 6\right)}{-\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \]
(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))
↓
(FPCore (x)
 :precision binary64
 (/ (fma -6.0 x 6.0) (- (fma 4.0 (sqrt x) (+ x 1.0)))))
double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}
↓
double code(double x) {
	return fma(-6.0, x, 6.0) / -fma(4.0, sqrt(x), (x + 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 Float64(fma(-6.0, x, 6.0) / Float64(-fma(4.0, sqrt(x), Float64(x + 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[(N[(-6.0 * x + 6.0), $MachinePrecision] / (-N[(4.0 * N[Sqrt[x], $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

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

Error

Bits error versus x

Target

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

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. Applied add-cube-cbrt_binary641.5

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(6, x, -6\right)} \cdot \sqrt[3]{\mathsf{fma}\left(6, x, -6\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(6, x, -6\right)}}}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)} \]

  4. Applied frac-2neg_binary641.5

    \[\leadsto \color{blue}{\frac{-\left(\sqrt[3]{\mathsf{fma}\left(6, x, -6\right)} \cdot \sqrt[3]{\mathsf{fma}\left(6, x, -6\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(6, x, -6\right)}}{-\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2022131 
(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)))))