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

Average Error: 0.4 → 0.3

Time: 2.7s

Precision: binary64

Math

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

↓

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

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))) (sqrt (/ -0.1111111111111111 (- x))))
  (* (sqrt x) -3.0)))

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))) + sqrt((-0.1111111111111111 / -x))) + (sqrt(x) * -3.0);
}

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))) + sqrt(((-0.1111111111111111d0) / -x))) + (sqrt(x) * (-3.0d0))
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))) + Math.sqrt((-0.1111111111111111 / -x))) + (Math.sqrt(x) * -3.0);
}

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))) + math.sqrt((-0.1111111111111111 / -x))) + (math.sqrt(x) * -3.0)

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(3.0 * Float64(y * sqrt(x))) + sqrt(Float64(-0.1111111111111111 / Float64(-x)))) + Float64(sqrt(x) * -3.0))
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))) + sqrt((-0.1111111111111111 / -x))) + (sqrt(x) * -3.0);
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[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(-0.1111111111111111 / (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.3

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

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 egg-rr 0.3

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022153 
(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)))