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

Average Accuracy: 99.9% → 99.8%

Time: 9.6s

Precision: binary64

Cost: 1600

Math

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

↓

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

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) (+ (+ 1.0 (* x (* x 0.04481))) -1.0))))
   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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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) + ((1.0d0 + (x * (x * 0.04481d0))) + (-1.0d0))))) - 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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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(Float64(x * 0.99229) + Float64(Float64(1.0 + Float64(x * Float64(x * 0.04481))) + -1.0)))) - 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) + ((1.0 + (x * (x * 0.04481))) + -1.0)))) - 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[(N[(x * 0.99229), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 99.8%

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

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	1216

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

Alternative 2
Accuracy	98.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{-82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.70711 \cdot \left(x \cdot \left(x \cdot 2.003561459544073 + -3.0191289437\right)\right) + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 3
Accuracy	98.3%
Cost	960

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

Alternative 4
Accuracy	98.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq -5.4:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{-82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 5
Accuracy	98.7%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 6
Accuracy	98.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 7
Accuracy	98.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 8
Accuracy	2.4%
Cost	64

\[-0.3135931908666891 \]

Alternative 9
Accuracy	51.2%
Cost	64

\[1.6316775383 \]

Reproduce

herbie shell --seed 2023141 
(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)))