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

Average Error: 0.1 → 0.2

Time: 5.6s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

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 * (x * x)));
}

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 * x)))
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 * (x * x)));
}

Python

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

↓

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

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 * Float64(x * x))))
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 * x)));
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[(x * x), $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 0.2

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

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	35.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1.5334083333333333\\ \end{array} \]

Alternative 2
Error	1.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot -0.253\\ \end{array} \]

Alternative 3
Error	1.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;1.5334083333333333 - 0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 + x \cdot -0.253\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \]

Alternative 4
Error	0.1
Cost	576

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

Alternative 5
Error	1.9
Cost	448

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

Alternative 6
Error	55.1
Cost	64

\[1.5334083333333333 \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))