Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 13.0s

Alternatives: 11

Speedup: 2.4×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a - \frac{1}{3}\\ t\_0 \cdot \left(1 + \frac{1}{\sqrt{9 \cdot t\_0}} \cdot rand\right) \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- a (/ 1.0 3.0))))
   (* t_0 (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 t_0))) rand)))))

C

double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    t_0 = a - (1.0d0 / 3.0d0)
    code = t_0 * (1.0d0 + ((1.0d0 / sqrt((9.0d0 * t_0))) * rand))
end function

Java

public static double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / Math.sqrt((9.0 * t_0))) * rand));
}

Python

def code(a, rand):
	t_0 = a - (1.0 / 3.0)
	return t_0 * (1.0 + ((1.0 / math.sqrt((9.0 * t_0))) * rand))

Julia

function code(a, rand)
	t_0 = Float64(a - Float64(1.0 / 3.0))
	return Float64(t_0 * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * t_0))) * rand)))
end

MATLAB

function tmp = code(a, rand)
	t_0 = a - (1.0 / 3.0);
	tmp = t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a - \frac{1}{3}\\ t\_0 \cdot \left(1 + \frac{1}{\sqrt{9 \cdot t\_0}} \cdot rand\right) \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- a (/ 1.0 3.0))))
   (* t_0 (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 t_0))) rand)))))

C

double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    t_0 = a - (1.0d0 / 3.0d0)
    code = t_0 * (1.0d0 + ((1.0d0 / sqrt((9.0d0 * t_0))) * rand))
end function

Java

public static double code(double a, double rand) {
	double t_0 = a - (1.0 / 3.0);
	return t_0 * (1.0 + ((1.0 / Math.sqrt((9.0 * t_0))) * rand));
}

Python

def code(a, rand):
	t_0 = a - (1.0 / 3.0)
	return t_0 * (1.0 + ((1.0 / math.sqrt((9.0 * t_0))) * rand))

Julia

function code(a, rand)
	t_0 = Float64(a - Float64(1.0 / 3.0))
	return Float64(t_0 * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * t_0))) * rand)))
end

MATLAB

function tmp = code(a, rand)
	t_0 = a - (1.0 / 3.0);
	tmp = t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (* (+ a -0.3333333333333333) (+ 1.0 (/ rand (sqrt (fma 9.0 a -3.0))))))

C

double code(double a, double rand) {
	return (a + -0.3333333333333333) * (1.0 + (rand / sqrt(fma(9.0, a, -3.0))));
}

Julia

function code(a, rand)
	return Float64(Float64(a + -0.3333333333333333) * Float64(1.0 + Float64(rand / sqrt(fma(9.0, a, -3.0)))))
end

Wolfram

code[a_, rand_] := N[(N[(a + -0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(rand / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval 99.8

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. *-lft-identity 99.9

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-neg N/A

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

   15. distribute-lft-in N/A

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

   16. lift-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 N/A

       \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \left(1 + \frac{rand}{\sqrt{\color{blue}{\mathsf{fma}\left(9, a, \mathsf{neg}\left(3\right)\right)}}}\right) \]

   22. metadata-eval 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}\right)} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333}\\ \mathbf{if}\;rand \leq -1.6 \cdot 10^{+68}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot t\_0\right)\\ \mathbf{elif}\;rand \leq 1.8 \cdot 10^{+63}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (sqrt (+ a -0.3333333333333333))))
   (if (<= rand -1.6e+68)
     (* rand (* 0.3333333333333333 t_0))
     (if (<= rand 1.8e+63)
       (+ a -0.3333333333333333)
       (* t_0 (* rand 0.3333333333333333))))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333));
	double tmp;
	if (rand <= -1.6e+68) {
		tmp = rand * (0.3333333333333333 * t_0);
	} else if (rand <= 1.8e+63) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (rand * 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a + (-0.3333333333333333d0)))
    if (rand <= (-1.6d+68)) then
        tmp = rand * (0.3333333333333333d0 * t_0)
    else if (rand <= 1.8d+63) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * (rand * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a + -0.3333333333333333));
	double tmp;
	if (rand <= -1.6e+68) {
		tmp = rand * (0.3333333333333333 * t_0);
	} else if (rand <= 1.8e+63) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (rand * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333))
	tmp = 0
	if rand <= -1.6e+68:
		tmp = rand * (0.3333333333333333 * t_0)
	elif rand <= 1.8e+63:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (rand * 0.3333333333333333)
	return tmp

Julia

function code(a, rand)
	t_0 = sqrt(Float64(a + -0.3333333333333333))
	tmp = 0.0
	if (rand <= -1.6e+68)
		tmp = Float64(rand * Float64(0.3333333333333333 * t_0));
	elseif (rand <= 1.8e+63)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(rand * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a + -0.3333333333333333));
	tmp = 0.0;
	if (rand <= -1.6e+68)
		tmp = rand * (0.3333333333333333 * t_0);
	elseif (rand <= 1.8e+63)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (rand * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[rand, -1.6e+68], N[(rand * N[(0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.8e+63], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.59999999999999997e68

   1. Initial program 99.4%

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

   3. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{rand \cdot \left(\left(\frac{1}{3} \cdot \sqrt{a - \frac{1}{3}} + \frac{a}{rand}\right) - \frac{1}{3} \cdot \frac{1}{rand}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto rand \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, \sqrt{a - \frac{1}{3}}, \frac{a - \frac{1}{3}}{rand}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto rand \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\sqrt{a - \frac{1}{3}}}, \frac{a - \frac{1}{3}}{rand}\right) \]

      8. sub-neg N/A

         \[\leadsto rand \cdot \mathsf{fma}\left(\frac{1}{3}, \sqrt{\color{blue}{a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}, \frac{a - \frac{1}{3}}{rand}\right) \]

      9. metadata-eval N/A

         \[\leadsto rand \cdot \mathsf{fma}\left(\frac{1}{3}, \sqrt{a + \color{blue}{\frac{-1}{3}}}, \frac{a - \frac{1}{3}}{rand}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto rand \cdot \mathsf{fma}\left(\frac{1}{3}, \sqrt{\color{blue}{a + \frac{-1}{3}}}, \frac{a - \frac{1}{3}}{rand}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto rand \cdot \mathsf{fma}\left(\frac{1}{3}, \sqrt{a + \frac{-1}{3}}, \color{blue}{\frac{a - \frac{1}{3}}{rand}}\right) \]

      12. sub-neg N/A

          \[\leadsto rand \cdot \mathsf{fma}\left(\frac{1}{3}, \sqrt{a + \frac{-1}{3}}, \frac{\color{blue}{a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}}{rand}\right) \]

      13. metadata-eval N/A

          \[\leadsto rand \cdot \mathsf{fma}\left(\frac{1}{3}, \sqrt{a + \frac{-1}{3}}, \frac{a + \color{blue}{\frac{-1}{3}}}{rand}\right) \]

      14. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in rand around inf

      \[\leadsto rand \cdot \left(\frac{1}{3} \cdot \color{blue}{\sqrt{a - \frac{1}{3}}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

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

   if -1.59999999999999997e68 < rand < 1.79999999999999999e63

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \color{blue}{\frac{-1}{3}} \]

      3. lower-+.f64 95.8

         \[\leadsto \color{blue}{a + -0.3333333333333333} \]

   5. Applied rewrites 95.8%

      \[\leadsto \color{blue}{a + -0.3333333333333333} \]

   if 1.79999999999999999e63 < rand

   1. Initial program 99.4%

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

   3. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. sub-neg N/A

         \[\leadsto \sqrt{\color{blue}{a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}} \cdot \left(\frac{1}{3} \cdot rand\right) \]

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\sqrt{a + -0.3333333333333333} \cdot \left(0.3333333333333333 \cdot rand\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.6 \cdot 10^{+68}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a + -0.3333333333333333}\right)\\ \mathbf{elif}\;rand \leq 1.8 \cdot 10^{+63}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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))))