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

Average Accuracy: 99.4% → 99.5%

Time: 9.8s

Precision: binary64

Cost: 7232

Math

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

↓

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

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

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 * 9.0)) * ((y + (1.0 / (x * 9.0))) + -1.0);
}

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 * 9.0d0)) * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))
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 * 9.0)) * ((y + (1.0 / (x * 9.0))) + -1.0);
}

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

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(Float64(x * 9.0)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
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 * 9.0)) * ((y + (1.0 / (x * 9.0))) + -1.0);
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[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Target

Original	99.4%
Target	99.4%
Herbie	99.5%

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

Derivation

1. Initial program 99.4%

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

2. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   Proof

   [Start] 99.5	\[ {\left(x \cdot 9\right)}^{0.5} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   unpow1/2 [=>] 99.5	\[ \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

4. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	85.0%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-67}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	85.0%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-67}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 3
Accuracy	58.4%
Cost	7117

\[\begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-67} \lor \neg \left(x \leq 7.5 \cdot 10^{-27}\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 4
Accuracy	58.4%
Cost	7116

\[\begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-67}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{0.3333333333333333}{y}}\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	7104

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

Alternative 6
Accuracy	99.4%
Cost	7104

\[\sqrt{x \cdot 9} \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) + -1\right) \]

Alternative 7
Accuracy	39.3%
Cost	6592

\[\sqrt{\frac{0.1111111111111111}{x}} \]

Reproduce

herbie shell --seed 2023122 
(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)))