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

Average Accuracy: 99.9% → 99.9%

Time: 5.9s

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 99.9%

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

2. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
   clear-num [=>] 99.9	\[ x - \color{blue}{\frac{1}{\frac{1 + \frac{x \cdot y}{2}}{y}}} \]
   inv-pow [=>] 99.9	\[ x - \color{blue}{{\left(\frac{1 + \frac{x \cdot y}{2}}{y}\right)}^{-1}} \]
   *-un-lft-identity [=>] 99.9	\[ x - {\left(\frac{\color{blue}{1 \cdot \left(1 + \frac{x \cdot y}{2}\right)}}{y}\right)}^{-1} \]
   *-un-lft-identity [<=] 99.9	\[ x - {\left(\frac{\color{blue}{1 + \frac{x \cdot y}{2}}}{y}\right)}^{-1} \]
   +-commutative [=>] 99.9	\[ x - {\left(\frac{\color{blue}{\frac{x \cdot y}{2} + 1}}{y}\right)}^{-1} \]
   div-inv [=>] 99.9	\[ x - {\left(\frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}} + 1}{y}\right)}^{-1} \]
   *-commutative [=>] 99.9	\[ x - {\left(\frac{\color{blue}{\left(y \cdot x\right)} \cdot \frac{1}{2} + 1}{y}\right)}^{-1} \]
   associate-*l* [=>] 99.9	\[ x - {\left(\frac{\color{blue}{y \cdot \left(x \cdot \frac{1}{2}\right)} + 1}{y}\right)}^{-1} \]
   fma-def [=>] 99.9	\[ x - {\left(\frac{\color{blue}{\mathsf{fma}\left(y, x \cdot \frac{1}{2}, 1\right)}}{y}\right)}^{-1} \]
   metadata-eval [=>] 99.9	\[ x - {\left(\frac{\mathsf{fma}\left(y, x \cdot \color{blue}{0.5}, 1\right)}{y}\right)}^{-1} \]

3. Taylor expanded in y around 0 99.9%

   \[\leadsto x - {\color{blue}{\left(\frac{1}{y} + 0.5 \cdot x\right)}}^{-1} \]

4. Applied egg-rr 99.9%

   \[\leadsto x - \color{blue}{\left(0 + \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}\right)} \]
   Proof

   [Start] 99.9	\[ x - {\left(\frac{1}{y} + 0.5 \cdot x\right)}^{-1} \]
   add-log-exp [=>] 63.7	\[ x - \color{blue}{\log \left(e^{{\left(\frac{1}{y} + 0.5 \cdot x\right)}^{-1}}\right)} \]
   *-un-lft-identity [=>] 63.7	\[ x - \log \color{blue}{\left(1 \cdot e^{{\left(\frac{1}{y} + 0.5 \cdot x\right)}^{-1}}\right)} \]
   log-prod [=>] 63.7	\[ x - \color{blue}{\left(\log 1 + \log \left(e^{{\left(\frac{1}{y} + 0.5 \cdot x\right)}^{-1}}\right)\right)} \]
   metadata-eval [=>] 63.7	\[ x - \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{y} + 0.5 \cdot x\right)}^{-1}}\right)\right) \]
   add-log-exp [<=] 99.9	\[ x - \left(0 + \color{blue}{{\left(\frac{1}{y} + 0.5 \cdot x\right)}^{-1}}\right) \]
   unpow-1 [=>] 99.9	\[ x - \left(0 + \color{blue}{\frac{1}{\frac{1}{y} + 0.5 \cdot x}}\right) \]
   +-commutative [=>] 99.9	\[ x - \left(0 + \frac{1}{\color{blue}{0.5 \cdot x + \frac{1}{y}}}\right) \]
   fma-def [=>] 99.9	\[ x - \left(0 + \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}}\right) \]

5. Simplified 99.9%

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

   [Start] 99.9	\[ x - \left(0 + \frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}\right) \]
   +-lft-identity [=>] 99.9	\[ x - \color{blue}{\frac{1}{\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)}} \]
   fma-udef [=>] 99.9	\[ x - \frac{1}{\color{blue}{0.5 \cdot x + \frac{1}{y}}} \]
   *-commutative [=>] 99.9	\[ x - \frac{1}{\color{blue}{x \cdot 0.5} + \frac{1}{y}} \]
   fma-udef [<=] 99.9	\[ x - \frac{1}{\color{blue}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)}} \]

6. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ x - \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1}{y}\right)} \]
   fma-udef [=>] 99.9	\[ x - \frac{1}{\color{blue}{x \cdot 0.5 + \frac{1}{y}}} \]
   +-commutative [=>] 99.9	\[ x - \frac{1}{\color{blue}{\frac{1}{y} + x \cdot 0.5}} \]

7. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	89.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+84}:\\ \;\;\;\;x + \frac{y \cdot \frac{-2}{y}}{x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+25}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-2}{x}\\ \end{array} \]

Alternative 2
Accuracy	89.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+83} \lor \neg \left(y \leq 3.7 \cdot 10^{+25}\right):\\ \;\;\;\;x + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 3
Accuracy	74.7%
Cost	192

\[x - y \]

Reproduce

herbie shell --seed 2023136 
(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)))))