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

Average Accuracy: 99.7% → 99.6%

Time: 9.7s

Precision: binary64

Cost: 7360

Math

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

↓

\[1 + \left(\frac{-1}{x \cdot 9} - \frac{\frac{y}{-{x}^{0.5}}}{-3}\right) \]

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 (- (pow x 0.5))) -3.0))))

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 + ((-1.0 / (x * 9.0)) - ((y / -pow(x, 0.5)) / -3.0));
}

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 + (((-1.0d0) / (x * 9.0d0)) - ((y / -(x ** 0.5d0)) / (-3.0d0)))
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 + ((-1.0 / (x * 9.0)) - ((y / -Math.pow(x, 0.5)) / -3.0));
}

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 + ((-1.0 / (x * 9.0)) - ((y / -math.pow(x, 0.5)) / -3.0))

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(1.0 + Float64(Float64(-1.0 / Float64(x * 9.0)) - Float64(Float64(y / Float64(-(x ^ 0.5))) / -3.0)))
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 + ((-1.0 / (x * 9.0)) - ((y / -(x ^ 0.5)) / -3.0));
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[(1.0 + N[(N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y / (-N[Power[x, 0.5], $MachinePrecision])), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	99.7%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Initial program 99.7%

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

2. Simplified 99.7%

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

   [Start] 99.7	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   sub-neg [=>] 99.7	\[ \color{blue}{\left(1 + \left(-\frac{1}{x \cdot 9}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   +-commutative [=>] 99.7	\[ \color{blue}{\left(\left(-\frac{1}{x \cdot 9}\right) + 1\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   associate--l+ [=>] 99.7	\[ \color{blue}{\left(-\frac{1}{x \cdot 9}\right) + \left(1 - \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   +-commutative [=>] 99.7	\[ \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)} \]
   associate-+l- [=>] 99.7	\[ \color{blue}{1 - \left(\frac{y}{3 \cdot \sqrt{x}} - \left(-\frac{1}{x \cdot 9}\right)\right)} \]
   sub-neg [=>] 99.7	\[ 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \left(-\left(-\frac{1}{x \cdot 9}\right)\right)\right)} \]
   +-commutative [=>] 99.7	\[ 1 - \color{blue}{\left(\left(-\left(-\frac{1}{x \cdot 9}\right)\right) + \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   remove-double-neg [<=] 99.7	\[ 1 - \left(\left(-\left(-\frac{1}{x \cdot 9}\right)\right) + \color{blue}{\left(-\left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]
   distribute-neg-in [<=] 99.7	\[ 1 - \color{blue}{\left(-\left(\left(-\frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)\right)} \]
   distribute-neg-in [<=] 99.7	\[ 1 - \left(-\color{blue}{\left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)}\right) \]
   remove-double-neg [=>] 99.7	\[ 1 - \color{blue}{\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]
   associate-/r* [=>] 99.7	\[ 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\right) \]

3. Applied egg-rr 99.5%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{1}{{x}^{0.25}} \cdot \frac{y}{\frac{{x}^{0.25}}{0.3333333333333333}}}\right) \]

4. Simplified 99.5%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y}{{x}^{0.25}} \cdot \frac{0.3333333333333333}{{x}^{0.25}}}\right) \]
   Proof

   [Start] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{1}{{x}^{0.25}} \cdot \frac{y}{\frac{{x}^{0.25}}{0.3333333333333333}}\right) \]
   associate-*l/ [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{1 \cdot \frac{y}{\frac{{x}^{0.25}}{0.3333333333333333}}}{{x}^{0.25}}}\right) \]
   *-lft-identity [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{\frac{y}{\frac{{x}^{0.25}}{0.3333333333333333}}}}{{x}^{0.25}}\right) \]
   associate-/r/ [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{\frac{y}{{x}^{0.25}} \cdot 0.3333333333333333}}{{x}^{0.25}}\right) \]
   associate-*r/ [<=] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y}{{x}^{0.25}} \cdot \frac{0.3333333333333333}{{x}^{0.25}}}\right) \]

5. Applied egg-rr 99.5%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{-1}{\frac{{x}^{0.25}}{y} \cdot \left({x}^{0.25} \cdot -3\right)}}\right) \]

6. Simplified 99.6%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{\frac{y}{-{x}^{0.5}}}{-3}}\right) \]
   Proof

   [Start] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{-1}{\frac{{x}^{0.25}}{y} \cdot \left({x}^{0.25} \cdot -3\right)}\right) \]
   associate-/r* [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{\frac{-1}{\frac{{x}^{0.25}}{y}}}{{x}^{0.25} \cdot -3}}\right) \]
   associate-/r* [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{\frac{\frac{-1}{\frac{{x}^{0.25}}{y}}}{{x}^{0.25}}}{-3}}\right) \]
   associate-/l* [<=] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\color{blue}{\frac{-1 \cdot y}{{x}^{0.25}}}}{{x}^{0.25}}}{-3}\right) \]
   neg-mul-1 [<=] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\frac{\color{blue}{-y}}{{x}^{0.25}}}{{x}^{0.25}}}{-3}\right) \]
   distribute-frac-neg [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\color{blue}{-\frac{y}{{x}^{0.25}}}}{{x}^{0.25}}}{-3}\right) \]
   neg-mul-1 [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\color{blue}{-1 \cdot \frac{y}{{x}^{0.25}}}}{{x}^{0.25}}}{-3}\right) \]
   metadata-eval [<=] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\color{blue}{\frac{1}{-1}} \cdot \frac{y}{{x}^{0.25}}}{{x}^{0.25}}}{-3}\right) \]
   times-frac [<=] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\color{blue}{\frac{1 \cdot y}{-1 \cdot {x}^{0.25}}}}{{x}^{0.25}}}{-3}\right) \]
   neg-mul-1 [<=] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\frac{1 \cdot y}{\color{blue}{-{x}^{0.25}}}}{{x}^{0.25}}}{-3}\right) \]
   associate-*r/ [<=] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\color{blue}{1 \cdot \frac{y}{-{x}^{0.25}}}}{{x}^{0.25}}}{-3}\right) \]
   *-lft-identity [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{\color{blue}{\frac{y}{-{x}^{0.25}}}}{{x}^{0.25}}}{-3}\right) \]
   associate-/l/ [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{\frac{y}{{x}^{0.25} \cdot \left(-{x}^{0.25}\right)}}}{-3}\right) \]
   distribute-rgt-neg-out [=>] 99.5	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{y}{\color{blue}{-{x}^{0.25} \cdot {x}^{0.25}}}}{-3}\right) \]
   pow-sqr [=>] 99.6	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{y}{-\color{blue}{{x}^{\left(2 \cdot 0.25\right)}}}}{-3}\right) \]
   metadata-eval [=>] 99.6	\[ 1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{y}{-{x}^{\color{blue}{0.5}}}}{-3}\right) \]

7. Final simplification 99.6%

   \[\leadsto 1 + \left(\frac{-1}{x \cdot 9} - \frac{\frac{y}{-{x}^{0.5}}}{-3}\right) \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7232

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

Alternative 2
Accuracy	94.8%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;y \leq -6.9 \cdot 10^{+53}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot {\left(x \cdot 9\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	7168

\[\left(1 + \frac{-0.1111111111111111}{x}\right) - y \cdot {\left(x \cdot 9\right)}^{-0.5} \]

Alternative 4
Accuracy	94.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+55} \lor \neg \left(y \leq 9.2 \cdot 10^{+34}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]

Alternative 5
Accuracy	98.5%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot {\left(x \cdot 9\right)}^{-0.5}\\ \end{array} \]

Alternative 6
Accuracy	99.6%
Cost	7104

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

Alternative 7
Accuracy	99.6%
Cost	7104

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

Alternative 8
Accuracy	92.0%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 9
Accuracy	92.0%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 10
Accuracy	92.1%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \left(\sqrt{\frac{1}{x}} \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+90}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 11
Accuracy	92.0%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+94} \lor \neg \left(y \leq 3.8 \cdot 10^{+90}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 12
Accuracy	92.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+90}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 13
Accuracy	92.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+90}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 14
Accuracy	92.1%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+90}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{x}}{-0.3333333333333333}}\\ \end{array} \]

Alternative 15
Accuracy	92.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -7.3 \cdot 10^{+93}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{x}}{-0.3333333333333333}}\\ \end{array} \]

Alternative 16
Accuracy	64.4%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 0.0105:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	65.5%
Cost	320

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

Alternative 18
Accuracy	33.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023141 
(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)))))