Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Average Error: 0.4 → 0.4

Time: 10.8s

Precision: binary64

Cost: 7104

Math

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

↓

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

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

↓

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (+ (/ 0.3333333333333333 x) (+ -3.0 (* y 3.0)))))

C

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 sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * ((0.3333333333333333d0 / x) + ((-3.0d0) + (y * 3.0d0)))
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

↓

public static double code(double x, double y) {
	return Math.sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)));
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

↓

def code(x, y):
	return math.sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)))

Julia

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 Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + Float64(-3.0 + Float64(y * 3.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

↓

function tmp = code(x, y)
	tmp = sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)));
end

Wolfram

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[(0.3333333333333333 / x), $MachinePrecision] + N[(-3.0 + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.4
Target	0.4
Herbie	0.4

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

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. Simplified 0.5

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

   [Start] 0.4	\[ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   *-commutative [=>] 0.4	\[ \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   associate-*l* [=>] 0.4	\[ \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]
   sub-neg [=>] 0.4	\[ \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]
   distribute-rgt-in [=>] 0.4	\[ \sqrt{x} \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) \cdot 3 + \left(-1\right) \cdot 3\right)} \]
   *-commutative [=>] 0.4	\[ \sqrt{x} \cdot \left(\color{blue}{3 \cdot \left(y + \frac{1}{x \cdot 9}\right)} + \left(-1\right) \cdot 3\right) \]
   fma-def [=>] 0.4	\[ \sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(3, y + \frac{1}{x \cdot 9}, \left(-1\right) \cdot 3\right)} \]
   *-commutative [=>] 0.4	\[ \sqrt{x} \cdot \mathsf{fma}\left(3, y + \frac{1}{\color{blue}{9 \cdot x}}, \left(-1\right) \cdot 3\right) \]
   associate-/r* [=>] 0.5	\[ \sqrt{x} \cdot \mathsf{fma}\left(3, y + \color{blue}{\frac{\frac{1}{9}}{x}}, \left(-1\right) \cdot 3\right) \]
   metadata-eval [=>] 0.5	\[ \sqrt{x} \cdot \mathsf{fma}\left(3, y + \frac{\color{blue}{0.1111111111111111}}{x}, \left(-1\right) \cdot 3\right) \]
   metadata-eval [=>] 0.5	\[ \sqrt{x} \cdot \mathsf{fma}\left(3, y + \frac{0.1111111111111111}{x}, \color{blue}{-1} \cdot 3\right) \]
   metadata-eval [=>] 0.5	\[ \sqrt{x} \cdot \mathsf{fma}\left(3, y + \frac{0.1111111111111111}{x}, \color{blue}{-3}\right) \]

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	9.8
Cost	7244

\[\begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \leq 12.8:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + y \cdot 3\right)\\ \end{array} \]

Alternative 2
Error	9.8
Cost	7244

\[\begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{3}{\frac{\sqrt{\frac{1}{x}}}{y}}\\ \mathbf{elif}\;x \leq 18:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + y \cdot 3\right)\\ \end{array} \]

Alternative 3
Error	11.2
Cost	7244

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{x}}{0.3333333333333333 \cdot \frac{1}{y}}\\ \mathbf{elif}\;x \leq 14:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + y \cdot 3\right)\\ \end{array} \]

Alternative 4
Error	1.1
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + \frac{0.1111111111111111}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 5
Error	1.2
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -1.08 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + \frac{0.1111111111111111}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 6
Error	1.2
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 7
Error	1.2
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -0.85 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{1}{\frac{x}{0.3333333333333333}}\right)\\ \end{array} \]

Alternative 8
Error	24.1
Cost	7117

\[\begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{elif}\;x \leq 44000000000000 \lor \neg \left(x \leq 2.3 \cdot 10^{+206}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 9
Error	24.1
Cost	7117

\[\begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 44000000000000 \lor \neg \left(x \leq 2.5 \cdot 10^{+206}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 10
Error	24.2
Cost	7117

\[\begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 48000000000000 \lor \neg \left(x \leq 2.6 \cdot 10^{+206}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 11
Error	24.1
Cost	7117

\[\begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 44000000000000 \lor \neg \left(x \leq 1.35 \cdot 10^{+206}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 12
Error	9.8
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+57} \lor \neg \left(y \leq 7 \cdot 10^{+82}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 13
Error	0.4
Cost	7104

\[3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \]

Alternative 14
Error	21.7
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 15
Error	61.9
Cost	6592

\[\sqrt{x \cdot 9} \]

Alternative 16
Error	46.2
Cost	6592

\[\sqrt{x} \cdot -3 \]

Reproduce

herbie shell --seed 2023034 
(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)))