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

Average Accuracy: 99.4% → 99.4%

Time: 12.2s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (+ (* 3.0 (* (sqrt x) (+ y -1.0))) (/ 0.3333333333333333 (sqrt x))))

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 (3.0 * (sqrt(x) * (y + -1.0))) + (0.3333333333333333 / sqrt(x));
}

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 = (3.0d0 * (sqrt(x) * (y + (-1.0d0)))) + (0.3333333333333333d0 / sqrt(x))
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 (3.0 * (Math.sqrt(x) * (y + -1.0))) + (0.3333333333333333 / Math.sqrt(x));
}

Python

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

↓

def code(x, y):
	return (3.0 * (math.sqrt(x) * (y + -1.0))) + (0.3333333333333333 / math.sqrt(x))

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(Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0))) + Float64(0.3333333333333333 / sqrt(x)))
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 = (3.0 * (sqrt(x) * (y + -1.0))) + (0.3333333333333333 / sqrt(x));
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[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	99.4%
Target	99.4%
Herbie	99.4%

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

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. Simplified 99.4%

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

   [Start] 99.4	\[ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   +-commutative [=>] 99.4	\[ \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} - 1\right) \]
   associate--l+ [=>] 99.4	\[ \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(y - 1\right)\right)} \]
   distribute-rgt-in [=>] 99.4	\[ \color{blue}{\frac{1}{x \cdot 9} \cdot \left(3 \cdot \sqrt{x}\right) + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)} \]
   remove-double-neg [<=] 99.4	\[ \color{blue}{\left(-\left(-\frac{1}{x \cdot 9} \cdot \left(3 \cdot \sqrt{x}\right)\right)\right)} + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   distribute-lft-neg-in [=>] 99.4	\[ \left(-\color{blue}{\left(-\frac{1}{x \cdot 9}\right) \cdot \left(3 \cdot \sqrt{x}\right)}\right) + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   distribute-rgt-neg-in [=>] 99.4	\[ \color{blue}{\left(-\frac{1}{x \cdot 9}\right) \cdot \left(-3 \cdot \sqrt{x}\right)} + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   mul-1-neg [<=] 99.4	\[ \left(-\frac{1}{x \cdot 9}\right) \cdot \color{blue}{\left(-1 \cdot \left(3 \cdot \sqrt{x}\right)\right)} + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   metadata-eval [<=] 99.4	\[ \left(-\frac{1}{x \cdot 9}\right) \cdot \left(\color{blue}{\left(-1\right)} \cdot \left(3 \cdot \sqrt{x}\right)\right) + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   *-commutative [=>] 99.4	\[ \left(-\frac{1}{\color{blue}{9 \cdot x}}\right) \cdot \left(\left(-1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   associate-/r* [=>] 99.3	\[ \left(-\color{blue}{\frac{\frac{1}{9}}{x}}\right) \cdot \left(\left(-1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   distribute-neg-frac [=>] 99.3	\[ \color{blue}{\frac{-\frac{1}{9}}{x}} \cdot \left(\left(-1\right) \cdot \left(3 \cdot \sqrt{x}\right)\right) + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   *-commutative [<=] 99.3	\[ \frac{-\frac{1}{9}}{x} \cdot \color{blue}{\left(\left(3 \cdot \sqrt{x}\right) \cdot \left(-1\right)\right)} + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   associate-/r/ [<=] 99.4	\[ \color{blue}{\frac{-\frac{1}{9}}{\frac{x}{\left(3 \cdot \sqrt{x}\right) \cdot \left(-1\right)}}} + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   associate-/l/ [<=] 99.4	\[ \frac{-\frac{1}{9}}{\color{blue}{\frac{\frac{x}{-1}}{3 \cdot \sqrt{x}}}} + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]
   associate-/r/ [=>] 99.3	\[ \color{blue}{\frac{-\frac{1}{9}}{\frac{x}{-1}} \cdot \left(3 \cdot \sqrt{x}\right)} + \left(y - 1\right) \cdot \left(3 \cdot \sqrt{x}\right) \]

3. Applied egg-rr 99.3%

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

   [Start] 99.4	\[ \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   add-sqr-sqrt [=>] 99.0	\[ \color{blue}{\left(\sqrt{3 \cdot \sqrt{x}} \cdot \sqrt{3 \cdot \sqrt{x}}\right)} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   sqrt-unprod [=>] 99.2	\[ \color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot \left(3 \cdot \sqrt{x}\right)}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   swap-sqr [=>] 99.1	\[ \sqrt{\color{blue}{\left(3 \cdot 3\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   metadata-eval [=>] 99.1	\[ \sqrt{\color{blue}{9} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   add-sqr-sqrt [<=] 99.3	\[ \sqrt{9 \cdot \color{blue}{x}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   *-commutative [=>] 99.3	\[ \sqrt{\color{blue}{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

4. Applied egg-rr 69.1%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot 9\right)}^{1.5}}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   Proof

   [Start] 99.3	\[ \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   add-cbrt-cube [=>] 69.1	\[ \color{blue}{\sqrt[3]{\left(\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}\right) \cdot \sqrt{x \cdot 9}}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   add-sqr-sqrt [<=] 69.1	\[ \sqrt[3]{\color{blue}{\left(x \cdot 9\right)} \cdot \sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   pow1 [=>] 69.1	\[ \sqrt[3]{\color{blue}{{\left(x \cdot 9\right)}^{1}} \cdot \sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   pow1/2 [=>] 69.1	\[ \sqrt[3]{{\left(x \cdot 9\right)}^{1} \cdot \color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   pow-prod-up [=>] 69.1	\[ \sqrt[3]{\color{blue}{{\left(x \cdot 9\right)}^{\left(1 + 0.5\right)}}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   metadata-eval [=>] 69.1	\[ \sqrt[3]{{\left(x \cdot 9\right)}^{\color{blue}{1.5}}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

5. Applied egg-rr 99.2%

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

   [Start] 69.1	\[ \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   distribute-rgt-in [=>] 69.1	\[ \color{blue}{\frac{0.1111111111111111}{x} \cdot \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} + \left(y - 1\right) \cdot \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}}} \]
   *-commutative [=>] 69.1	\[ \frac{0.1111111111111111}{x} \cdot \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} + \color{blue}{\sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} \cdot \left(y - 1\right)} \]
   add-sqr-sqrt [=>] 69.1	\[ \color{blue}{\left(\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}\right)} \cdot \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} + \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} \cdot \left(y - 1\right) \]
   associate-*l* [=>] 69.1	\[ \color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \left(\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}}\right)} + \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} \cdot \left(y - 1\right) \]
   fma-def [=>] 69.1	\[ \color{blue}{\mathsf{fma}\left(\sqrt{\frac{0.1111111111111111}{x}}, \sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}}, \sqrt[3]{{\left(x \cdot 9\right)}^{1.5}} \cdot \left(y - 1\right)\right)} \]

6. Simplified 99.4%

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

   [Start] 99.2	\[ \mathsf{fma}\left(\frac{0.3333333333333333}{\sqrt{x}}, \frac{0.3333333333333333}{\sqrt{x}} \cdot \left(3 \cdot \sqrt{x}\right), \left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\right) \]
   fma-udef [=>] 99.2	\[ \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot \left(\frac{0.3333333333333333}{\sqrt{x}} \cdot \left(3 \cdot \sqrt{x}\right)\right) + \left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)} \]
   +-commutative [=>] 99.2	\[ \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \left(\frac{0.3333333333333333}{\sqrt{x}} \cdot \left(3 \cdot \sqrt{x}\right)\right)} \]
   associate-*l* [=>] 99.2	\[ \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)} + \frac{0.3333333333333333}{\sqrt{x}} \cdot \left(\frac{0.3333333333333333}{\sqrt{x}} \cdot \left(3 \cdot \sqrt{x}\right)\right) \]
   *-commutative [<=] 99.2	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.3333333333333333}{\sqrt{x}}\right)} \]
   associate-*r/ [=>] 99.3	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.3333333333333333}{\sqrt{x}}} \]
   *-commutative [<=] 99.3	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.3333333333333333}{\sqrt{x}} \]
   associate-*l* [=>] 99.4	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.3333333333333333\right)}}{\sqrt{x}} \]
   metadata-eval [=>] 99.4	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\sqrt{x} \cdot \color{blue}{1}}{\sqrt{x}} \]
   *-rgt-identity [=>] 99.4	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\color{blue}{\sqrt{x}}}{\sqrt{x}} \]
   *-inverses [=>] 99.4	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \frac{0.3333333333333333}{\sqrt{x}} \cdot \color{blue}{1} \]
   *-rgt-identity [=>] 99.4	\[ 3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right) + \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	84.2%
Cost	7376

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+51}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+60}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 2
Accuracy	64.3%
Cost	7249

\[\begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 92000000 \lor \neg \left(x \leq 2.8 \cdot 10^{+226}\right) \land x \leq 1.18 \cdot 10^{+265}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 3
Accuracy	64.2%
Cost	7249

\[\begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 33000000 \lor \neg \left(x \leq 5.7 \cdot 10^{+224}\right) \land x \leq 2.22 \cdot 10^{+266}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	7232

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

Alternative 5
Accuracy	99.4%
Cost	7104

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

Alternative 6
Accuracy	99.4%
Cost	7104

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

Alternative 7
Accuracy	58.2%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -30 \lor \neg \left(y \leq 3.2 \cdot 10^{-16}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 8
Accuracy	84.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 3.25 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 9
Accuracy	3.3%
Cost	6592

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

Alternative 10
Accuracy	26.4%
Cost	6592

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

Reproduce

herbie shell --seed 2023151 
(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)))