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

Average Error: 0.1 → 0.1

Time: 5.3s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 * 0.253d0) + (0.12d0 * (x ** 2.0d0)))
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 - ((x * 0.253) + (0.12 * Math.pow(x, 2.0)));
}

Python

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

↓

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

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 * 0.253) + Float64(0.12 * (x ^ 2.0))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 1.0 - ((x * 0.253) + (0.12 * (x ^ 2.0)));
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[(x * 0.253), $MachinePrecision] + N[(0.12 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Taylor expanded in x around 0 0.1

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

3. Simplified 0.1

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

   [Start] 0.1	\[ 1 - \left(0.253 \cdot x + 0.12 \cdot {x}^{2}\right) \]
   rational.json-simplify-2 [=>] 0.1	\[ 1 - \left(\color{blue}{x \cdot 0.253} + 0.12 \cdot {x}^{2}\right) \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	704

\[1 - x \cdot \left(\left(x \cdot 0.12 - -1.253\right) + -1\right) \]

Alternative 2
Error	0.1
Cost	576

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

Alternative 3
Error	2.0
Cost	448

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

Alternative 4
Error	21.1
Cost	320

\[1 - x \cdot 0.253 \]

Alternative 5
Error	22.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))