Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Average Error: 0.1 → 0.1

Time: 7.0s

Precision: binary64

Cost: 1216

Math

\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

↓

\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

↓

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

↓

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

↓

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

↓

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

↓

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

↓

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

2. Final simplification 0.1

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	1096

\[\begin{array}{l} t_0 := 0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} + \frac{\frac{-82.23527511657367}{x}}{x}\right) - x\right)\\ \mathbf{if}\;x \leq -524.5358328543535:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.39211478482209317:\\ \;\;\;\;x \cdot -2.134856267379707 + \left(1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.6
Cost	968

\[\begin{array}{l} t_0 := \frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{if}\;x \leq -524.5358328543535:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.39211478482209317:\\ \;\;\;\;x \cdot -2.134856267379707 + \left(1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.0
Cost	960

\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} - x\right) \]

Alternative 4
Error	0.7
Cost	840

\[\begin{array}{l} t_0 := \frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{if}\;x \leq -524.5358328543535:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.39211478482209317:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.7
Cost	712

\[\begin{array}{l} t_0 := \frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{if}\;x \leq -524.5358328543535:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.39211478482209317:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -524.5358328543535:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.39211478482209317:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 7
Error	1.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -524.5358328543535:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.39211478482209317:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 8
Error	31.5
Cost	64

\[1.6316775383 \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))