Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B

Average Error: 0.06% → 0.06%

Time: 5.2s

Precision: binary64

Cost: 704

Math

\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]

↓

\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]

FPCore

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))

↓

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))

C

double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

↓

double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + ((x * y) / 2.0d0)))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (y / (1.0d0 + ((x * y) / 2.0d0)))
end function

Java

public static double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

↓

public static double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}

Python

def code(x, y):
	return x - (y / (1.0 + ((x * y) / 2.0)))

↓

def code(x, y):
	return x - (y / (1.0 + ((x * y) / 2.0)))

Julia

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end

↓

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (y / (1.0 + ((x * y) / 2.0)));
end

↓

function tmp = code(x, y)
	tmp = x - (y / (1.0 + ((x * y) / 2.0)));
end

Wolfram

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.06

   \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]

2. Final simplification 0.06

   \[\leadsto x - \frac{y}{1 + \frac{x \cdot y}{2}} \]

Alternatives

Alternative 1
Error	0.09%
Cost	704

\[x - \frac{y}{1 + \frac{x}{\frac{2}{y}}} \]

Alternative 2
Error	14.56%
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00115:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	8.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+118} \lor \neg \left(y \leq 5.2 \cdot 10^{+62}\right):\\ \;\;\;\;x + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 4
Error	12.77%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -14800:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	21.64%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-113}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	36.82%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))