Average Error: 0.4 → 0.3
Time: 2.5s
Precision: binary64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
↓
\[\mathsf{fma}\left(\sqrt{x}, 3 \cdot y + -3, \sqrt{\frac{0.1111111111111111}{x}}\right) \]
(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) (+ (* 3.0 y) -3.0) (sqrt (/ 0.1111111111111111 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), ((3.0 * y) + -3.0), sqrt((0.1111111111111111 / 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), Float64(Float64(3.0 * y) + -3.0), sqrt(Float64(0.1111111111111111 / 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[(N[(3.0 * y), $MachinePrecision] + -3.0), $MachinePrecision] + N[Sqrt[N[(0.1111111111111111 / x), $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}, 3 \cdot y + -3, \sqrt{\frac{0.1111111111111111}{x}}\right)

Error

Bits error versus x

Bits error versus y

Target

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

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. Simplified0.4

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

  3. Taylor expanded in y around 0 0.4

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

  4. Simplified0.4

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

  5. Applied egg-rr0.3

    \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, -3\right), \color{blue}{\sqrt{\frac{1}{x} \cdot 0.1111111111111111}}\right) \]

  6. Taylor expanded in x around 0 0.3

    \[\leadsto \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, -3\right), \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}}\right) \]

  7. Taylor expanded in y around 0 0.3

    \[\leadsto \mathsf{fma}\left(\sqrt{x}, \color{blue}{3 \cdot y - 3}, \sqrt{\frac{0.1111111111111111}{x}}\right) \]

  8. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\sqrt{x}, 3 \cdot y + -3, \sqrt{\frac{0.1111111111111111}{x}}\right) \]

Reproduce

herbie shell --seed 2022166 
(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)))