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

Average Error: 0.1 → 0.1

Time: 2.1s

Precision: binary64

Math

\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

↓

\[1 + \left({x}^{2} \cdot -0.12 + x \cdot -0.253\right) \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

↓

(FPCore (x) :precision binary64 (+ 1.0 (+ (* (pow x 2.0) -0.12) (* x -0.253))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

↓

double code(double x) {
	return 1.0 + ((pow(x, 2.0) * -0.12) + (x * -0.253));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (((x ** 2.0d0) * (-0.12d0)) + (x * (-0.253d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

↓

public static double code(double x) {
	return 1.0 + ((Math.pow(x, 2.0) * -0.12) + (x * -0.253));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

↓

def code(x):
	return 1.0 + ((math.pow(x, 2.0) * -0.12) + (x * -0.253))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

↓

function code(x)
	return Float64(1.0 + Float64(Float64((x ^ 2.0) * -0.12) + Float64(x * -0.253)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

↓

function tmp = code(x)
	tmp = 1.0 + (((x ^ 2.0) * -0.12) + (x * -0.253));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(1.0 + N[(N[(N[Power[x, 2.0], $MachinePrecision] * -0.12), $MachinePrecision] + N[(x * -0.253), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Derivation

1. Initial program 0.1

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{1 - \left(0.12 \cdot {x}^{2} + 0.253 \cdot x\right)} \]

4. Final simplification 0.1

   \[\leadsto 1 + \left({x}^{2} \cdot -0.12 + x \cdot -0.253\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))