Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.1

Time: 10.7s

Precision: binary64

Cost: 7232

Math

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

↓

\[\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{a \cdot 9 + -3}}\right) \]

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) (+ 1.0 (/ rand (sqrt (+ (* a 9.0) -3.0))))))

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) * (1.0 + (rand / sqrt(((a * 9.0) + -3.0))));
}

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)) * (1.0d0 + (rand / sqrt(((a * 9.0d0) + (-3.0d0)))))
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) * (1.0 + (rand / Math.sqrt(((a * 9.0) + -3.0))));
}

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) * (1.0 + (rand / math.sqrt(((a * 9.0) + -3.0))))

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 + -0.3333333333333333) * Float64(1.0 + Float64(rand / sqrt(Float64(Float64(a * 9.0) + -3.0)))))
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) * (1.0 + (rand / sqrt(((a * 9.0) + -3.0))));
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 + -0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(rand / N[Sqrt[N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{a \cdot 9 + -3}}\right)} \]
   Proof
   (*.f64 (+.f64 a -1/3) (+.f64 1 (/.f64 rand (sqrt.f64 (+.f64 (*.f64 a 9) -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 a (Rewrite<= metadata-eval (neg.f64 1/3))) (+.f64 1 (/.f64 rand (sqrt.f64 (+.f64 (*.f64 a 9) -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 a (neg.f64 (Rewrite<= metadata-eval (/.f64 1 3)))) (+.f64 1 (/.f64 rand (sqrt.f64 (+.f64 (*.f64 a 9) -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 a (/.f64 1 3))) (+.f64 1 (/.f64 rand (sqrt.f64 (+.f64 (*.f64 a 9) -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 rand)) (sqrt.f64 (+.f64 (*.f64 a 9) -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (+.f64 (*.f64 a 9) (Rewrite<= metadata-eval (*.f64 -1/3 9))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (+.f64 (*.f64 a 9) (*.f64 (Rewrite<= metadata-eval (neg.f64 1/3)) 9)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (+.f64 (*.f64 a 9) (*.f64 (neg.f64 (Rewrite<= metadata-eval (/.f64 1 3))) 9)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 9 (+.f64 a (neg.f64 (/.f64 1 3))))))))): 1 points increase in error, 1 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (*.f64 9 (Rewrite<= sub-neg_binary64 (-.f64 a (/.f64 1 3)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)))): 11 points increase in error, 6 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))))): 0 points increase in error, 2 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)))): 2 points increase in error, 0 points decrease in error

3. Final simplification 0.1

   \[\leadsto \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{a \cdot 9 + -3}}\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	7232

\[\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{\frac{rand}{\sqrt{a + -0.3333333333333333}}}{3}\right) \]

Alternative 2
Error	5.5
Cost	7112

\[\begin{array}{l} t_0 := rand \cdot \frac{\sqrt{a + -0.3333333333333333}}{3}\\ \mathbf{if}\;rand \leq -6 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 9.2 \cdot 10^{+63}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	5.5
Cost	7112

\[\begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333}\\ \mathbf{if}\;rand \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;rand \cdot \frac{t_0}{3}\\ \mathbf{elif}\;rand \leq 1.12 \cdot 10^{+64}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{3}{rand}}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7104

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

Alternative 5
Error	0.2
Cost	7104

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

Alternative 6
Error	18.8
Cost	192

\[a + -0.3333333333333333 \]

Alternative 7
Error	63.1
Cost	64

\[-0.3333333333333333 \]

Alternative 8
Error	19.5
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022296 
(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))))