Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Average Error: 0.32% → 0.31%

Time: 8.3s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (- 1.0 (+ (/ 1.0 (* x 9.0)) (* (sqrt (/ 0.1111111111111111 x)) y))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
}

↓

double code(double x, double y) {
	return 1.0 - ((1.0 / (x * 9.0)) + (sqrt((0.1111111111111111 / x)) * y));
}

Fortran

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - ((1.0d0 / (x * 9.0d0)) + (sqrt((0.1111111111111111d0 / x)) * y))
end function

Java

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

↓

public static double code(double x, double y) {
	return 1.0 - ((1.0 / (x * 9.0)) + (Math.sqrt((0.1111111111111111 / x)) * y));
}

Python

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

↓

def code(x, y):
	return 1.0 - ((1.0 / (x * 9.0)) + (math.sqrt((0.1111111111111111 / x)) * y))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

↓

function code(x, y)
	return Float64(1.0 - Float64(Float64(1.0 / Float64(x * 9.0)) + Float64(sqrt(Float64(0.1111111111111111 / x)) * y)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = 1.0 - ((1.0 / (x * 9.0)) + (sqrt((0.1111111111111111 / x)) * y));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(1.0 - N[(N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.32%
Target	0.34%
Herbie	0.31%

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

Derivation

1. Initial program 0.32

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

2. Simplified 0.32

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

   [Start] 0.32	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   sub-neg [=>] 0.32	\[ \color{blue}{\left(1 + \left(-\frac{1}{x \cdot 9}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   +-commutative [=>] 0.32	\[ \color{blue}{\left(\left(-\frac{1}{x \cdot 9}\right) + 1\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   associate--l+ [=>] 0.32	\[ \color{blue}{\left(-\frac{1}{x \cdot 9}\right) + \left(1 - \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   +-commutative [=>] 0.32	\[ \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)} \]
   associate-+l- [=>] 0.32	\[ \color{blue}{1 - \left(\frac{y}{3 \cdot \sqrt{x}} - \left(-\frac{1}{x \cdot 9}\right)\right)} \]
   sub-neg [=>] 0.32	\[ 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \left(-\left(-\frac{1}{x \cdot 9}\right)\right)\right)} \]
   +-commutative [=>] 0.32	\[ 1 - \color{blue}{\left(\left(-\left(-\frac{1}{x \cdot 9}\right)\right) + \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   remove-double-neg [<=] 0.32	\[ 1 - \left(\left(-\left(-\frac{1}{x \cdot 9}\right)\right) + \color{blue}{\left(-\left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]
   distribute-neg-in [<=] 0.32	\[ 1 - \color{blue}{\left(-\left(\left(-\frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right)} \]
   distribute-neg-in [<=] 0.32	\[ 1 - \left(-\color{blue}{\left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]
   remove-double-neg [=>] 0.32	\[ 1 - \color{blue}{\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   associate-/r* [=>] 0.32	\[ 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\right) \]

3. Applied egg-rr 0.31

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

4. Final simplification 0.31

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

Alternatives

Alternative 1
Error	6.4%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -28000000000000 \lor \neg \left(y \leq 4.5 \cdot 10^{+26}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]

Alternative 2
Error	8.32%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+70}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+50}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]

Alternative 3
Error	0.36%
Cost	7104

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

Alternative 4
Error	8.36%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+70} \lor \neg \left(y \leq 7 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]

Alternative 5
Error	8.47%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+68} \lor \neg \left(y \leq 6.6 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 6
Error	34.67%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	33.47%
Cost	320

\[1 + \frac{-0.1111111111111111}{x} \]

Alternative 8
Error	66.15%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))