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

Average Error: 0.1 → 0.2

Time: 3.6s

Precision: binary64

Cost: 1088

Math

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

↓

\[1 - x \cdot \frac{0.064009 + x \cdot \left(x \cdot -0.0144\right)}{0.253 + x \cdot -0.12} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (- 1.0 (* x (/ (+ 0.064009 (* x (* x -0.0144))) (+ 0.253 (* x -0.12))))))

C

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

↓

double code(double x) {
	return 1.0 - (x * ((0.064009 + (x * (x * -0.0144))) / (0.253 + (x * -0.12))));
}

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 * ((0.064009d0 + (x * (x * (-0.0144d0)))) / (0.253d0 + (x * (-0.12d0)))))
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 * ((0.064009 + (x * (x * -0.0144))) / (0.253 + (x * -0.12))));
}

Python

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

↓

def code(x):
	return 1.0 - (x * ((0.064009 + (x * (x * -0.0144))) / (0.253 + (x * -0.12))))

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(x * Float64(Float64(0.064009 + Float64(x * Float64(x * -0.0144))) / Float64(0.253 + Float64(x * -0.12)))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 1.0 - (x * ((0.064009 + (x * (x * -0.0144))) / (0.253 + (x * -0.12))));
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[(x * N[(N[(0.064009 + N[(x * N[(x * -0.0144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.253 + N[(x * -0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.3

   \[\leadsto 1 - x \cdot \color{blue}{\left(\frac{0.064009}{0.253 + -0.12 \cdot x} - \frac{\left(x \cdot x\right) \cdot 0.0144}{0.253 + -0.12 \cdot x}\right)} \]

3. Simplified 0.2

   \[\leadsto 1 - x \cdot \color{blue}{\frac{0.064009 - x \cdot \left(x \cdot 0.0144\right)}{0.253 + x \cdot -0.12}} \]
   Proof

   [Start] 0.3	\[ 1 - x \cdot \left(\frac{0.064009}{0.253 + -0.12 \cdot x} - \frac{\left(x \cdot x\right) \cdot 0.0144}{0.253 + -0.12 \cdot x}\right) \]
   div-sub [<=] 0.3	\[ 1 - x \cdot \color{blue}{\frac{0.064009 - \left(x \cdot x\right) \cdot 0.0144}{0.253 + -0.12 \cdot x}} \]
   associate-*l* [=>] 0.2	\[ 1 - x \cdot \frac{0.064009 - \color{blue}{x \cdot \left(x \cdot 0.0144\right)}}{0.253 + -0.12 \cdot x} \]
   *-commutative [=>] 0.2	\[ 1 - x \cdot \frac{0.064009 - x \cdot \left(x \cdot 0.0144\right)}{0.253 + \color{blue}{x \cdot -0.12}} \]

4. Final simplification 0.2

   \[\leadsto 1 - x \cdot \frac{0.064009 + x \cdot \left(x \cdot -0.0144\right)}{0.253 + x \cdot -0.12} \]

Alternatives

Alternative 1
Error	0.1
Cost	704

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

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	21.0
Cost	320

\[1 + x \cdot -0.253 \]

Alternative 5
Error	22.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))