Average Error: 0.4 → 0.4
Time: 3.3s
Precision: binary64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
↓
\[\mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, -3\right), 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) \]
(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))
↓
(FPCore (x y)
 :precision binary64
 (fma (sqrt x) (fma 3.0 y -3.0) (* 0.3333333333333333 (sqrt (/ 1.0 x)))))
double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}
↓
double code(double x, double y) {
	return fma(sqrt(x), fma(3.0, y, -3.0), (0.3333333333333333 * sqrt((1.0 / x))));
}

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end
↓
function code(x, y)
	return fma(sqrt(x), fma(3.0, y, -3.0), Float64(0.3333333333333333 * sqrt(Float64(1.0 / x))))
end

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision] + N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, -3\right), 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)

Error

Bits error versus x

Bits error versus y

Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right) \]

Derivation

  1. Initial program 0.4

    \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

  2. Taylor expanded in y around 0 0.4

    \[\leadsto \color{blue}{\left(3 \cdot \left(y \cdot \sqrt{x}\right) + 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) - 3 \cdot \sqrt{x}} \]

  3. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, -3\right), 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)} \]

  4. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, -3\right), 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))