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

Average Error: 0.0 → 0.0

Time: 6.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (- x (* y (/ 1.0 (+ 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 / (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 / (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 / (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 / (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(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 / (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[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.1
Cost	704

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

Alternative 2
Error	0.0
Cost	704

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

Alternative 3
Error	0.0
Cost	704

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

Alternative 4
Error	5.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+67} \lor \neg \left(y \leq 6 \cdot 10^{+165}\right):\\ \;\;\;\;x + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 5
Error	8.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -850000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-28}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	13.3
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-115}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	23.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(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)))))