Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D

Average Error: 0.0 → 0.0

Time: 7.4s

Precision: binary64

Cost: 1984

Math

\[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\\ x - \left(t_0 \cdot \left(x \cdot 0.27061\right) + t_0 \cdot 2.30753\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))))
   (- x (+ (* t_0 (* x 0.27061)) (* t_0 2.30753)))))

C

double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

↓

double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * (0.99229 + (x * 0.04481))));
	return x - ((t_0 * (x * 0.27061)) + (t_0 * 2.30753));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + ((0.99229d0 + (x * 0.04481d0)) * x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))
    code = x - ((t_0 * (x * 0.27061d0)) + (t_0 * 2.30753d0))
end function

Java

public static double code(double x) {
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
}

↓

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (x * (0.99229 + (x * 0.04481))));
	return x - ((t_0 * (x * 0.27061)) + (t_0 * 2.30753));
}

Python

def code(x):
	return x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)))

↓

def code(x):
	t_0 = 1.0 / (1.0 + (x * (0.99229 + (x * 0.04481))))
	return x - ((t_0 * (x * 0.27061)) + (t_0 * 2.30753))

Julia

function code(x)
	return Float64(x - Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(Float64(0.99229 + Float64(x * 0.04481)) * x))))
end

↓

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481)))))
	return Float64(x - Float64(Float64(t_0 * Float64(x * 0.27061)) + Float64(t_0 * 2.30753)))
end

MATLAB

function tmp = code(x)
	tmp = x - ((2.30753 + (x * 0.27061)) / (1.0 + ((0.99229 + (x * 0.04481)) * x)));
end

↓

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (x * (0.99229 + (x * 0.04481))));
	tmp = x - ((t_0 * (x * 0.27061)) + (t_0 * 2.30753));
end

Wolfram

code[x_] := N[(x - N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x - N[(N[(t$95$0 * N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 2.30753), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.0

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]

2. Applied egg-rr 0.0

   \[\leadsto x - \color{blue}{\left(\frac{1}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \cdot \left(x \cdot 0.27061\right) + \frac{1}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \cdot 2.30753\right)} \]

3. Final simplification 0.0

   \[\leadsto x - \left(\frac{1}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot \left(x \cdot 0.27061\right) + \frac{1}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot 2.30753\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	1088

\[x + \frac{-2.30753 - x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \]

Alternative 2
Error	1.0
Cost	832

\[x + \frac{-2.30753 - x \cdot 0.27061}{1 + x \cdot 0.99229} \]

Alternative 3
Error	1.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	1.5
Cost	192

\[x + -2.30753 \]

Alternative 5
Error	31.7
Cost	64

\[-2.30753 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))