Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.2

Time: 8.4s

Precision: binary64

Cost: 7104

Math

\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

↓

\[\left(a + 0.3333333333333333 \cdot \left(\sqrt{a + -0.3333333333333333} \cdot rand\right)\right) + -0.3333333333333333 \]

FPCore

(FPCore (a rand)
 :precision binary64
 (*
  (- a (/ 1.0 3.0))
  (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))

↓

(FPCore (a rand)
 :precision binary64
 (+
  (+ a (* 0.3333333333333333 (* (sqrt (+ a -0.3333333333333333)) rand)))
  -0.3333333333333333))

C

double code(double a, double rand) {
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
}

↓

double code(double a, double rand) {
	return (a + (0.3333333333333333 * (sqrt((a + -0.3333333333333333)) * rand))) + -0.3333333333333333;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a - (1.0d0 / 3.0d0)) * (1.0d0 + ((1.0d0 / sqrt((9.0d0 * (a - (1.0d0 / 3.0d0))))) * rand))
end function

↓

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a + (0.3333333333333333d0 * (sqrt((a + (-0.3333333333333333d0))) * rand))) + (-0.3333333333333333d0)
end function

Java

public static double code(double a, double rand) {
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / Math.sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
}

↓

public static double code(double a, double rand) {
	return (a + (0.3333333333333333 * (Math.sqrt((a + -0.3333333333333333)) * rand))) + -0.3333333333333333;
}

Python

def code(a, rand):
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / math.sqrt((9.0 * (a - (1.0 / 3.0))))) * rand))

↓

def code(a, rand):
	return (a + (0.3333333333333333 * (math.sqrt((a + -0.3333333333333333)) * rand))) + -0.3333333333333333

Julia

function code(a, rand)
	return Float64(Float64(a - Float64(1.0 / 3.0)) * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * Float64(a - Float64(1.0 / 3.0))))) * rand)))
end

↓

function code(a, rand)
	return Float64(Float64(a + Float64(0.3333333333333333 * Float64(sqrt(Float64(a + -0.3333333333333333)) * rand))) + -0.3333333333333333)
end

MATLAB

function tmp = code(a, rand)
	tmp = (a - (1.0 / 3.0)) * (1.0 + ((1.0 / sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
end

↓

function tmp = code(a, rand)
	tmp = (a + (0.3333333333333333 * (sqrt((a + -0.3333333333333333)) * rand))) + -0.3333333333333333;
end

Wolfram

code[a_, rand_] := N[(N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, rand_] := N[(N[(a + N[(0.3333333333333333 * N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{1}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}} \cdot rand\right)} \]
   Proof
   (*.f64 (+.f64 a -1/3) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 (+.f64 a -1/3) 9))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 a (Rewrite<= metadata-eval (neg.f64 1/3))) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 (+.f64 a -1/3) 9))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 a (neg.f64 (Rewrite<= metadata-eval (/.f64 1 3)))) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 (+.f64 a -1/3) 9))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 a (/.f64 1 3))) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 (+.f64 a -1/3) 9))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 (+.f64 a (Rewrite<= metadata-eval (neg.f64 1/3))) 9))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 (+.f64 a (neg.f64 (Rewrite<= metadata-eval (/.f64 1 3)))) 9))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 a (/.f64 1 3))) 9))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 9 (-.f64 a (/.f64 1 3)))))) rand))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (Rewrite<= sub-neg_binary64 (-.f64 1 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1 (-.f64 a (/.f64 1 3))) (*.f64 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)) (-.f64 a (/.f64 1 3))))): 1 points increase in error, 3 points decrease in error
   (-.f64 (Rewrite=> *-lft-identity_binary64 (-.f64 a (/.f64 1 3))) (*.f64 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)) (-.f64 a (/.f64 1 3)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub_binary64 (+.f64 (-.f64 a (/.f64 1 3)) (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand) (-.f64 a (/.f64 1 3))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 a (/.f64 1 3)))) (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand) (-.f64 a (/.f64 1 3)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-in_binary64 (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)))): 3 points increase in error, 1 points decrease in error

3. Taylor expanded in rand around 0 0.2

   \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(\sqrt{a - 0.3333333333333333} \cdot rand\right) + a\right) - 0.3333333333333333} \]

4. Final simplification 0.2

   \[\leadsto \left(a + 0.3333333333333333 \cdot \left(\sqrt{a + -0.3333333333333333} \cdot rand\right)\right) + -0.3333333333333333 \]

Alternatives

Alternative 1
Error	5.5
Cost	7112

\[\begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(\sqrt{a + -0.3333333333333333} \cdot rand\right)\\ \mathbf{if}\;rand \leq -8.5 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	5.4
Cost	7112

\[\begin{array}{l} t_0 := rand \cdot \sqrt{\left(a + -0.3333333333333333\right) \cdot 0.1111111111111111}\\ \mathbf{if}\;rand \leq -4.2 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 2 \cdot 10^{+82}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7104

\[\left(a + -0.3333333333333333\right) \cdot \left(1 + 0.3333333333333333 \cdot \frac{rand}{\sqrt{a}}\right) \]

Alternative 4
Error	0.9
Cost	7104

\[\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{a \cdot 9}}\right) \]

Alternative 5
Error	5.9
Cost	6984

\[\begin{array}{l} t_0 := rand \cdot \sqrt{a \cdot 0.1111111111111111}\\ \mathbf{if}\;rand \leq -4.2 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 1.55 \cdot 10^{+82}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	18.2
Cost	192

\[a + -0.3333333333333333 \]

Alternative 7
Error	18.9
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022338 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))