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

Average Error: 0.2 → 0.3

Time: 3.9s

Precision: binary64

Cost: 7360

Math

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

↓

\[\left(1 - \frac{1}{x \cdot 9}\right) - y \cdot \left(0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) \]

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 (* 0.3333333333333333 (sqrt (/ 1.0 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 * (0.3333333333333333 * sqrt((1.0 / 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 * (0.3333333333333333d0 * sqrt((1.0d0 / 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 * (0.3333333333333333 * Math.sqrt((1.0 / 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 * (0.3333333333333333 * math.sqrt((1.0 / 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(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y * Float64(0.3333333333333333 * sqrt(Float64(1.0 / 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 * (0.3333333333333333 * sqrt((1.0 / 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[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 0.2

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

2. Taylor expanded in y around 0 0.3

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

3. Simplified 0.3

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

   [Start] 0.3	\[ \left(1 - \frac{1}{x \cdot 9}\right) - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.3	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{y \cdot \left(0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)} \]

4. Final simplification 0.3

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - y \cdot \left(0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	7232

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

Alternative 2
Error	0.2
Cost	7232

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

Alternative 3
Error	0.2
Cost	7104

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

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)))))