Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Average Error: 0.4 → 0.4

Time: 6.0s

Precision: binary64

Math

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

↓

\[\left(\left(3 \cdot y\right) \cdot \sqrt{x} + \frac{0.3333333333333333}{\sqrt{x}}\right) - 3 \cdot \sqrt{x} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

↓

(FPCore (x y)
 :precision binary64
 (-
  (+ (* (* 3.0 y) (sqrt x)) (/ 0.3333333333333333 (sqrt x)))
  (* 3.0 (sqrt x))))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

↓

double code(double x, double y) {
	return (((3.0 * y) * sqrt(x)) + (0.3333333333333333 / sqrt(x))) - (3.0 * sqrt(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((3.0d0 * y) * sqrt(x)) + (0.3333333333333333d0 / sqrt(x))) - (3.0d0 * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

↓

public static double code(double x, double y) {
	return (((3.0 * y) * Math.sqrt(x)) + (0.3333333333333333 / Math.sqrt(x))) - (3.0 * Math.sqrt(x));
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

↓

def code(x, y):
	return (((3.0 * y) * math.sqrt(x)) + (0.3333333333333333 / math.sqrt(x))) - (3.0 * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

↓

function code(x, y)
	return Float64(Float64(Float64(Float64(3.0 * y) * sqrt(x)) + Float64(0.3333333333333333 / sqrt(x))) - Float64(3.0 * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

↓

function tmp = code(x, y)
	tmp = (((3.0 * y) * sqrt(x)) + (0.3333333333333333 / sqrt(x))) - (3.0 * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 0.4

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

2. Taylor expanded in y around 0 0.4

   \[\leadsto \color{blue}{\left(3 \cdot \left(y \cdot \sqrt{x}\right) + 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) - 3 \cdot \sqrt{x}} \]

3. Applied sqrt-div_binary64 0.4

   \[\leadsto \left(3 \cdot \left(y \cdot \sqrt{x}\right) + 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) - 3 \cdot \sqrt{x} \]

4. Applied associate-*r/_binary64 0.4

   \[\leadsto \left(3 \cdot \left(y \cdot \sqrt{x}\right) + \color{blue}{\frac{0.3333333333333333 \cdot \sqrt{1}}{\sqrt{x}}}\right) - 3 \cdot \sqrt{x} \]

5. Simplified 0.4

   \[\leadsto \left(3 \cdot \left(y \cdot \sqrt{x}\right) + \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}}\right) - 3 \cdot \sqrt{x} \]

6. Applied associate-*r*_binary64 0.4

   \[\leadsto \left(\color{blue}{\left(3 \cdot y\right) \cdot \sqrt{x}} + \frac{0.3333333333333333}{\sqrt{x}}\right) - 3 \cdot \sqrt{x} \]

7. Final simplification 0.4

   \[\leadsto \left(\left(3 \cdot y\right) \cdot \sqrt{x} + \frac{0.3333333333333333}{\sqrt{x}}\right) - 3 \cdot \sqrt{x} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))