Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 9.1s

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 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   *-lft-identity [<=] 0.1	\[ \color{blue}{\left(1 \cdot \left(a - \frac{1}{3}\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   *-lft-identity [=>] 0.1	\[ \color{blue}{\left(a - \frac{1}{3}\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   sub-neg [=>] 0.1	\[ \color{blue}{\left(a + \left(-\frac{1}{3}\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + \left(-\color{blue}{0.3333333333333333}\right)\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + \color{blue}{-0.3333333333333333}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   cancel-sign-sub [<=] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \color{blue}{\left(1 - \left(-\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand\right)} \]
   distribute-lft-neg-in [<=] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 - \color{blue}{\left(-\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]
   sub-neg [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \color{blue}{\left(1 + \left(-\left(-\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right)\right)} \]
   remove-double-neg [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \color{blue}{\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand}\right) \]
   associate-*l/ [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \color{blue}{\frac{1 \cdot rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right) \]
   *-lft-identity [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{\color{blue}{rand}}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \]
   sub-neg [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{9 \cdot \color{blue}{\left(a + \left(-\frac{1}{3}\right)\right)}}}\right) \]
   distribute-lft-in [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\color{blue}{9 \cdot a + 9 \cdot \left(-\frac{1}{3}\right)}}}\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{9 \cdot a + 9 \cdot \left(-\color{blue}{0.3333333333333333}\right)}}\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{9 \cdot a + 9 \cdot \color{blue}{-0.3333333333333333}}}\right) \]
   metadata-eval [=>] 0.1	\[ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{9 \cdot a + \color{blue}{-3}}}\right) \]

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	5.8
Cost	7113

\[\begin{array}{l} \mathbf{if}\;rand \leq -4.3 \cdot 10^{+85} \lor \neg \left(rand \leq 3.6 \cdot 10^{+45}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a + -0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;a + -0.3333333333333333\\ \end{array} \]

Alternative 2
Error	5.7
Cost	7113

\[\begin{array}{l} \mathbf{if}\;rand \leq -6.8 \cdot 10^{+84} \lor \neg \left(rand \leq 2.65 \cdot 10^{+45}\right):\\ \;\;\;\;rand \cdot \sqrt{-0.037037037037037035 + a \cdot 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;a + -0.3333333333333333\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7104

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

Alternative 4
Error	0.2
Cost	7104

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

Alternative 5
Error	18.5
Cost	192

\[a + -0.3333333333333333 \]

Alternative 6
Error	19.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023039 
(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))))