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

Average Error: 0.0 → 0.1

Time: 9.8s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (+ x (/ -1.0 (+ (/ 1.0 y) (* x 0.5)))))

C

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

↓

double code(double x, double y) {
	return x + (-1.0 / ((1.0 / y) + (x * 0.5)));
}

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 + ((-1.0d0) / ((1.0d0 / y) + (x * 0.5d0)))
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 + (-1.0 / ((1.0 / y) + (x * 0.5)));
}

Python

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

↓

def code(x, y):
	return x + (-1.0 / ((1.0 / y) + (x * 0.5)))

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(-1.0 / Float64(Float64(1.0 / y) + Float64(x * 0.5))))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = x + (-1.0 / ((1.0 / y) + (x * 0.5)));
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[(-1.0 / N[(N[(1.0 / y), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.1

   \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}} \]
   Proof

   [Start] 0.0	\[ x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   sub-neg [=>] 0.0	\[ \color{blue}{x + \left(-\frac{y}{1 + \frac{x \cdot y}{2}}\right)} \]
   neg-mul-1 [=>] 0.0	\[ x + \color{blue}{-1 \cdot \frac{y}{1 + \frac{x \cdot y}{2}}} \]
   associate-*r/ [=>] 0.0	\[ x + \color{blue}{\frac{-1 \cdot y}{1 + \frac{x \cdot y}{2}}} \]
   associate-/l* [=>] 0.1	\[ x + \color{blue}{\frac{-1}{\frac{1 + \frac{x \cdot y}{2}}{y}}} \]
   associate-/l* [=>] 0.1	\[ x + \frac{-1}{\frac{1 + \color{blue}{\frac{x}{\frac{2}{y}}}}{y}} \]
   remove-double-neg [<=] 0.1	\[ x + \frac{-1}{\frac{1 + \frac{\color{blue}{-\left(-x\right)}}{\frac{2}{y}}}{y}} \]
   distribute-frac-neg [=>] 0.1	\[ x + \frac{-1}{\frac{1 + \color{blue}{\left(-\frac{-x}{\frac{2}{y}}\right)}}{y}} \]
   unsub-neg [=>] 0.1	\[ x + \frac{-1}{\frac{\color{blue}{1 - \frac{-x}{\frac{2}{y}}}}{y}} \]
   div-sub [=>] 0.1	\[ x + \frac{-1}{\color{blue}{\frac{1}{y} - \frac{\frac{-x}{\frac{2}{y}}}{y}}} \]
   associate-/r/ [=>] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\color{blue}{\frac{-x}{2} \cdot y}}{y}} \]
   associate-/l* [=>] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \color{blue}{\frac{\frac{-x}{2}}{\frac{y}{y}}}} \]
   remove-double-neg [<=] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2}}{\frac{y}{\color{blue}{-\left(-y\right)}}}} \]
   mul-1-neg [<=] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2}}{\frac{y}{\color{blue}{-1 \cdot \left(-y\right)}}}} \]
   associate-/r* [=>] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2}}{\color{blue}{\frac{\frac{y}{-1}}{-y}}}} \]
   associate-/l* [<=] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \color{blue}{\frac{\frac{-x}{2} \cdot \left(-y\right)}{\frac{y}{-1}}}} \]
   metadata-eval [<=] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2} \cdot \left(-y\right)}{\frac{y}{\color{blue}{\frac{1}{-1}}}}} \]
   associate-/l* [<=] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2} \cdot \left(-y\right)}{\color{blue}{\frac{y \cdot -1}{1}}}} \]
   *-commutative [<=] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2} \cdot \left(-y\right)}{\frac{\color{blue}{-1 \cdot y}}{1}}} \]
   neg-mul-1 [<=] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2} \cdot \left(-y\right)}{\frac{\color{blue}{-y}}{1}}} \]
   /-rgt-identity [=>] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2} \cdot \left(-y\right)}{\color{blue}{-y}}} \]
   associate-/l* [=>] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \color{blue}{\frac{\frac{-x}{2}}{\frac{-y}{-y}}}} \]
   *-inverses [=>] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \frac{\frac{-x}{2}}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 0.1	\[ x + \frac{-1}{\frac{1}{y} - \color{blue}{\frac{-x}{2}}} \]

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	5.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+92} \lor \neg \left(y \leq 2.3 \cdot 10^{+105}\right):\\ \;\;\;\;x + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 2
Error	8.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	24.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(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)))))