Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 20.0s

Precision: binary64

Cost: 7232

Math

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

↓

\[\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\frac{0.3333333333333333 - a}{-0.1111111111111111}}}\right) \]

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 (/ (- 0.3333333333333333 a) -0.1111111111111111))))))

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(((0.3333333333333333 - a) / -0.1111111111111111))));
}

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(((0.3333333333333333d0 - a) / (-0.1111111111111111d0)))))
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(((0.3333333333333333 - a) / -0.1111111111111111))));
}

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(((0.3333333333333333 - a) / -0.1111111111111111))))

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(0.3333333333333333 - a) / -0.1111111111111111)))))
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(((0.3333333333333333 - a) / -0.1111111111111111))));
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[(0.3333333333333333 - a), $MachinePrecision] / -0.1111111111111111), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

   [Start] 0.1	\[ \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-63 [<=] 0.1	\[ \color{blue}{\left(\left(a - \frac{1}{3}\right) \cdot 1\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-39 [<=] 0.1	\[ \color{blue}{\left(1 \cdot \left(a - \frac{1}{3}\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-25 [=>] 0.1	\[ \color{blue}{\left(\left(-1\right) \cdot \left(\frac{1}{3} - a\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-51 [=>] 0.2	\[ \left(\left(-1\right) \cdot \color{blue}{\left(\left(\frac{1}{3} + \frac{1}{3}\right) - \left(\frac{1}{3} + a\right)\right)}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-41 [<=] 0.2	\[ \left(\left(-1\right) \cdot \left(\left(\frac{1}{3} + \frac{1}{3}\right) - \color{blue}{\left(a + \frac{1}{3}\right)}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-25 [<=] 0.2	\[ \color{blue}{\left(1 \cdot \left(\left(a + \frac{1}{3}\right) - \left(\frac{1}{3} + \frac{1}{3}\right)\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-34 [=>] 0.2	\[ \color{blue}{\left(\left(a + \frac{1}{3}\right) \cdot 1 - 1 \cdot \left(\frac{1}{3} + \frac{1}{3}\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-63 [=>] 0.2	\[ \left(\color{blue}{\left(a + \frac{1}{3}\right)} - 1 \cdot \left(\frac{1}{3} + \frac{1}{3}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.2	\[ \left(\left(a + \frac{1}{3}\right) - 1 \cdot \left(\color{blue}{0.3333333333333333} + \frac{1}{3}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.2	\[ \left(\left(a + \frac{1}{3}\right) - 1 \cdot \left(0.3333333333333333 + \color{blue}{0.3333333333333333}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.2	\[ \left(\left(a + \frac{1}{3}\right) - 1 \cdot \color{blue}{0.6666666666666666}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.2	\[ \left(\left(a + \frac{1}{3}\right) - \color{blue}{0.6666666666666666}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [<=] 0.2	\[ \left(\left(a + \frac{1}{3}\right) - \color{blue}{\left(0.3333333333333333 + 0.3333333333333333\right)}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [<=] 0.2	\[ \left(\left(a + \frac{1}{3}\right) - \left(\color{blue}{\frac{1}{3}} + 0.3333333333333333\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [<=] 0.2	\[ \left(\left(a + \frac{1}{3}\right) - \left(\frac{1}{3} + \color{blue}{\frac{1}{3}}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   rational.json-simplify-13 [=>] 0.1	\[ \color{blue}{\left(a + \left(\frac{1}{3} - \left(\frac{1}{3} + \frac{1}{3}\right)\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + \left(\color{blue}{0.3333333333333333} - \left(\frac{1}{3} + \frac{1}{3}\right)\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + \left(0.3333333333333333 - \left(\color{blue}{0.3333333333333333} + \frac{1}{3}\right)\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + \left(0.3333333333333333 - \left(0.3333333333333333 + \color{blue}{0.3333333333333333}\right)\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + \left(0.3333333333333333 - \color{blue}{0.6666666666666666}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + \color{blue}{-0.3333333333333333}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

3. Applied egg-rr 0.1

   \[\leadsto \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\color{blue}{\frac{0.3333333333333333 - a}{-0.1111111111111111}}}}\right) \]

4. Final simplification 0.1

   \[\leadsto \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\frac{0.3333333333333333 - a}{-0.1111111111111111}}}\right) \]

Alternatives

Alternative 1
Error	5.7
Cost	7240

\[\begin{array}{l} t_0 := rand \cdot \left(\sqrt{\frac{0.1111111111111111}{a}} \cdot \left(a - 0.3333333333333333\right)\right)\\ \mathbf{if}\;rand \leq -6.8 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+78}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	5.0
Cost	7240

\[\begin{array}{l} t_0 := rand \cdot \left(\sqrt{\frac{0.1111111111111111}{a}} \cdot \left(a - 0.3333333333333333\right)\right)\\ \mathbf{if}\;rand \leq -2.8 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 8.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{a \cdot rand}{\sqrt{a \cdot 9}} + a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7232

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

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	18.4
Cost	192

\[a - 0.3333333333333333 \]

Alternative 6
Error	19.2
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023066 
(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))))