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

Average Error: 0.4 → 0.4

Time: 14.3s

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	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. Applied egg-rr 0.4

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

3. Simplified 0.4

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

   [Start] 0.4	\[ {\left(x \cdot 9\right)}^{0.5} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   unpow1/2 [=>] 0.4	\[ \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	9.5
Cost	7508

\[\begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ t_1 := t_0 \cdot y\\ \mathbf{if}\;x \leq 1.06 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7200:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 2
Error	9.5
Cost	7508

\[\begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ t_1 := t_0 \cdot y\\ \mathbf{if}\;x \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 3
Error	9.6
Cost	7508

\[\begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{3 \cdot \sqrt{x}}{\frac{1}{y}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;t_0 \cdot y\\ \mathbf{elif}\;x \leq 76000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 4
Error	22.0
Cost	7248

\[\begin{array}{l} t_0 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 4.1 \cdot 10^{-72}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 5
Error	22.0
Cost	7248

\[\begin{array}{l} t_0 := \sqrt{x \cdot 9} \cdot y\\ \mathbf{if}\;x \leq 2.2 \cdot 10^{-67}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 6
Error	22.0
Cost	7248

\[\begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ t_1 := t_0 \cdot y\\ \mathbf{if}\;x \leq 1.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 7
Error	9.4
Cost	7244

\[\begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ \mathbf{if}\;x \leq 1.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-55}:\\ \;\;\;\;t_0 \cdot y\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 8
Error	1.4
Cost	7244

\[\begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + \frac{0.1111111111111111}{x}\right)\right)\\ \mathbf{elif}\;x \leq 460000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\\ \mathbf{elif}\;x \leq 1500000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 9
Error	1.5
Cost	7244

\[\begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\left(y + \frac{0.1111111111111111}{x}\right) \cdot t_0\\ \mathbf{elif}\;x \leq 880000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\\ \mathbf{elif}\;x \leq 1500000:\\ \;\;\;\;y \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 10
Error	1.4
Cost	7244

\[\begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{0.5}}{0.3333333333333333} \cdot \left(y + \frac{0.1111111111111111}{x}\right)\\ \mathbf{elif}\;x \leq 196000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\\ \mathbf{elif}\;x \leq 1500000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 11
Error	1.5
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \lor \neg \left(y \leq 1.12 \cdot 10^{-18}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + \frac{0.1111111111111111}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 12
Error	0.4
Cost	7104

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

Alternative 13
Error	0.4
Cost	7104

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

Alternative 14
Error	22.1
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 15
Error	38.7
Cost	6592

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

Reproduce

herbie shell --seed 2023060 
(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)))