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

Average Error: 0.0 → 0.1

Time: 4.8s

Precision: binary64

Cost: 1024

Math

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

↓

\[\left(x + x\right) + \left(-\left(x + \frac{y}{1 + \frac{y \cdot x}{2}}\right)\right) \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (+ (+ x x) (- (+ x (/ y (+ 1.0 (/ (* y x) 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 + x) + -(x + (y / (1.0 + ((y * x) / 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 + x) + -(x + (y / (1.0d0 + ((y * x) / 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 + x) + -(x + (y / (1.0 + ((y * x) / 2.0))));
}

Python

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

↓

def code(x, y):
	return (x + x) + -(x + (y / (1.0 + ((y * x) / 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(Float64(x + x) + Float64(-Float64(x + Float64(y / Float64(1.0 + Float64(Float64(y * x) / 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 + x) + -(x + (y / (1.0 + ((y * x) / 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[(N[(x + x), $MachinePrecision] + (-N[(x + N[(y / N[(1.0 + N[(N[(y * x), $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 0.1

   \[\leadsto \color{blue}{\left(x + x\right) + \left(0 - \left(x + \frac{y}{1 + \frac{y \cdot x}{2}}\right)\right)} \]

3. Simplified 0.1

   \[\leadsto \color{blue}{\left(x + x\right) + \left(-\left(x + \frac{y}{1 + \frac{y \cdot x}{2}}\right)\right)} \]
   Proof

   [Start] 0.1	\[ \left(x + x\right) + \left(0 - \left(x + \frac{y}{1 + \frac{y \cdot x}{2}}\right)\right) \]
   rational_best.json-simplify-10 [=>] 0.1	\[ \left(x + x\right) + \color{blue}{\left(-\left(x + \frac{y}{1 + \frac{y \cdot x}{2}}\right)\right)} \]

4. Final simplification 0.1

   \[\leadsto \left(x + x\right) + \left(-\left(x + \frac{y}{1 + \frac{y \cdot x}{2}}\right)\right) \]

Alternatives

Alternative 1
Error	5.6
Cost	904

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+89}:\\ \;\;\;\;x - \frac{2}{x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+80}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;\left(x + x\right) + \left(-\left(x + \frac{2}{x}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.0
Cost	704

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

Alternative 3
Error	5.6
Cost	584

\[\begin{array}{l} t_0 := x - \frac{2}{x}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+79}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	8.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.65:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-9}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	13.8
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-126}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	23.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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)))))