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

Average Error: 0.2 → 0.2

Time: 11.5s

Precision: binary64

Cost: 7232

Math

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

↓

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

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)) (/ (/ y 3.0) (sqrt x)))))

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)) - ((y / 3.0) / sqrt(x)));
}

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)) - ((y / 3.0d0) / sqrt(x)))
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)) - ((y / 3.0) / Math.sqrt(x)));
}

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)) - ((y / 3.0) / math.sqrt(x)))

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(Float64(y / 3.0) / sqrt(x))))
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)) - ((y / 3.0) / sqrt(x)));
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[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

   [Start] 0.2	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   sub-neg [=>] 0.2	\[ \color{blue}{\left(1 + \left(-\frac{1}{x \cdot 9}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   +-commutative [=>] 0.2	\[ \color{blue}{\left(\left(-\frac{1}{x \cdot 9}\right) + 1\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   associate--l+ [=>] 0.2	\[ \color{blue}{\left(-\frac{1}{x \cdot 9}\right) + \left(1 - \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   +-commutative [=>] 0.2	\[ \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)} \]
   associate-+l- [=>] 0.2	\[ \color{blue}{1 - \left(\frac{y}{3 \cdot \sqrt{x}} - \left(-\frac{1}{x \cdot 9}\right)\right)} \]
   sub-neg [=>] 0.2	\[ 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \left(-\left(-\frac{1}{x \cdot 9}\right)\right)\right)} \]
   +-commutative [=>] 0.2	\[ 1 - \color{blue}{\left(\left(-\left(-\frac{1}{x \cdot 9}\right)\right) + \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   remove-double-neg [<=] 0.2	\[ 1 - \left(\left(-\left(-\frac{1}{x \cdot 9}\right)\right) + \color{blue}{\left(-\left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]
   distribute-neg-in [<=] 0.2	\[ 1 - \color{blue}{\left(-\left(\left(-\frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right)} \]
   distribute-neg-in [<=] 0.2	\[ 1 - \left(-\color{blue}{\left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]
   remove-double-neg [=>] 0.2	\[ 1 - \color{blue}{\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   associate-/r* [=>] 0.2	\[ 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\right) \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	3.3
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+46} \lor \neg \left(y \leq 6.5 \cdot 10^{+44}\right):\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 2
Error	1.2
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;x \leq 16200:\\ \;\;\;\;\frac{1 + \frac{\frac{-0.012345679012345678}{x}}{x}}{1 + \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 3
Error	1.2
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;x \leq 32000:\\ \;\;\;\;\frac{1 + \frac{\frac{-0.012345679012345678}{x}}{x}}{1 + \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 4
Error	3.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+46} \lor \neg \left(y \leq 6.2 \cdot 10^{+44}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 5
Error	0.2
Cost	7104

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

Alternative 6
Error	5.3
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+59} \lor \neg \left(y \leq 1.2 \cdot 10^{+47}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 7
Error	5.3
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+60} \lor \neg \left(y \leq 1.2 \cdot 10^{+47}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 8
Error	5.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+60}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+47}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]

Alternative 9
Error	5.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+47}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]

Alternative 10
Error	5.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+47}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]

Alternative 11
Error	5.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+58}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]

Alternative 12
Error	21.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	21.3
Cost	320

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

Alternative 14
Error	42.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(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)))))