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

Average Error: 0.35% → 0.39%

Time: 9.5s

Precision: binary64

Cost: 7104

Math

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

↓

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

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 (/ 0.1111111111111111 x)) (/ y (sqrt (* x 9.0)))))

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 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
}

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 - (0.1111111111111111d0 / x)) - (y / sqrt((x * 9.0d0)))
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 - (0.1111111111111111 / x)) - (y / Math.sqrt((x * 9.0)));
}

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 - (0.1111111111111111 / x)) - (y / math.sqrt((x * 9.0)))

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(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / sqrt(Float64(x * 9.0))))
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 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
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[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.35%
Target	0.35%
Herbie	0.39%

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

Derivation

1. Initial program 0.35

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

2. Simplified 0.38

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

   [Start] 0.35	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   sub-neg [=>] 0.35	\[ \color{blue}{\left(1 + \left(-\frac{1}{x \cdot 9}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   +-commutative [=>] 0.35	\[ \color{blue}{\left(\left(-\frac{1}{x \cdot 9}\right) + 1\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   neg-sub0 [=>] 0.35	\[ \left(\color{blue}{\left(0 - \frac{1}{x \cdot 9}\right)} + 1\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-+l- [=>] 0.35	\[ \color{blue}{\left(0 - \left(\frac{1}{x \cdot 9} - 1\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-+l- [<=] 0.35	\[ \color{blue}{\left(\left(0 - \frac{1}{x \cdot 9}\right) + 1\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   neg-sub0 [<=] 0.35	\[ \left(\color{blue}{\left(-\frac{1}{x \cdot 9}\right)} + 1\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   +-commutative [<=] 0.35	\[ \color{blue}{\left(1 + \left(-\frac{1}{x \cdot 9}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   distribute-neg-frac [=>] 0.35	\[ \left(1 + \color{blue}{\frac{-1}{x \cdot 9}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   *-commutative [=>] 0.35	\[ \left(1 + \frac{-1}{\color{blue}{9 \cdot x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-/r* [=>] 0.38	\[ \left(1 + \color{blue}{\frac{\frac{-1}{9}}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   metadata-eval [=>] 0.38	\[ \left(1 + \frac{\frac{\color{blue}{-1}}{9}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   metadata-eval [=>] 0.38	\[ \left(1 + \frac{\color{blue}{-0.1111111111111111}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

3. Applied egg-rr 0.39

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

4. Final simplification 0.39

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

Alternatives

Alternative 1
Error	5.14%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+41} \lor \neg \left(y \leq 9 \cdot 10^{+53}\right):\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 2
Error	5.13%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+42}:\\ \;\;\;\;1 + y \cdot \frac{-1}{\frac{\sqrt{x}}{0.3333333333333333}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 3
Error	5.16%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;1 + y \cdot \frac{-1}{\frac{\sqrt{x}}{0.3333333333333333}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+51}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{3}{\sqrt{x}} \cdot \left(-0.1111111111111111 \cdot y\right)\\ \end{array} \]

Alternative 4
Error	5.17%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+41}:\\ \;\;\;\;1 + y \cdot \frac{-1}{\frac{\sqrt{x}}{0.3333333333333333}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+49}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -3}{9 \cdot \sqrt{x}}\\ \end{array} \]

Alternative 5
Error	5.12%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+42} \lor \neg \left(y \leq 3.7 \cdot 10^{+50}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 6
Error	0.37%
Cost	7104

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

Alternative 7
Error	7.84%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+59} \lor \neg \left(y \leq 1.02 \cdot 10^{+89}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 8
Error	7.88%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+59} \lor \neg \left(y \leq 1.7 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 9
Error	7.9%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+59}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+91}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 10
Error	7.89%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{x}}{-0.3333333333333333}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+91}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]

Alternative 11
Error	34.23%
Cost	324

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

Alternative 12
Error	33.31%
Cost	320

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

Alternative 13
Error	66.12%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023088 
(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)))))