Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2

Average Error: 0.1 → 0.0

Time: 1.5s

Precision: binary64

Cost: 448

Math

\[\frac{x + y}{y + y} \]

↓

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

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (+ y y)))

↓

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

C

double code(double x, double y) {
	return (x + y) / (y + y);
}

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (y + y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 + ((0.5d0 * x) / y)
end function

Java

public static double code(double x, double y) {
	return (x + y) / (y + y);
}

↓

public static double code(double x, double y) {
	return 0.5 + ((0.5 * x) / y);
}

Python

def code(x, y):
	return (x + y) / (y + y)

↓

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

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + y))
end

↓

function code(x, y)
	return Float64(0.5 + Float64(Float64(0.5 * x) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / (y + y);
end

↓

function tmp = code(x, y)
	tmp = 0.5 + ((0.5 * x) / y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(0.5 + N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Target

Original	0.1
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.1

   \[\frac{x + y}{y + y} \]

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{0.5 + \frac{0.5 \cdot x}{y}} \]
   Proof
   (+.f64 1/2 (/.f64 (*.f64 1/2 x) y)): 0 points increase in error, 0 points decrease in error
   (+.f64 1/2 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 x y)))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	18.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 2
Error	62.5
Cost	64

\[-1 \]

Alternative 3
Error	27.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x y)
  :name "Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2"
  :precision binary64

  :herbie-target
  (+ (* 0.5 (/ x y)) 0.5)

  (/ (+ x y) (+ y y)))