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

Average Error: 0.2 → 0.2

Time: 7.1s

Precision: binary64

Cost: 7104

Math

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

↓

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

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)) (* (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 + (-0.1111111111111111 / x)) - (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 + ((-0.1111111111111111d0) / x)) - (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 + (-0.1111111111111111 / x)) - (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 + (-0.1111111111111111 / x)) - (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(Float64(1.0 + Float64(-0.1111111111111111 / x)) - 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 + (-0.1111111111111111 / x)) - (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[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision] * y), $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. Taylor expanded in x around 0 0.2

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

3. Applied egg-rr 0.3

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot y} \]

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	3.7
Cost	7176

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3239738512990265 \cdot 10^{+48}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 30230995247775824:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 2
Error	3.7
Cost	7112

\[\begin{array}{l} t_0 := 1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -2.3239738512990265 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 30230995247775824:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3239738512990265 \cdot 10^{+48}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 30230995247775824:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 4
Error	5.8
Cost	7048

\[\begin{array}{l} t_0 := -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{if}\;y \leq -8.718852606254243 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.7208773642170624 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	5.8
Cost	6984

\[\begin{array}{l} t_0 := \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -8.718852606254243 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.7208773642170624 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	5.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -8.718852606254243 \cdot 10^{+65}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.7208773642170624 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 7
Error	21.8
Cost	320

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

Alternative 8
Error	42.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022291 
(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)))))