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

Average Accuracy: 99.9% → 99.9%

Time: 6.0s

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 99.9%

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

2. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	85.3%
Cost	848

\[\begin{array}{l} t_0 := x + \frac{-2}{x}\\ \mathbf{if}\;x \leq -3 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-163}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	85.1%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -340000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-163}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	704

\[x + \frac{-1}{\frac{1}{y} + x \cdot 0.5} \]

Alternative 4
Accuracy	86.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -340000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	78.6%
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-117}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	62.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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)))))