Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 11.8s

Alternatives: 19

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (1.0d0 + (rand / sqrt((9.0d0 * (a + (-0.3333333333333333d0)))))) * (a + (-0.3333333333333333d0))
end function

Java

public static double code(double a, double rand) {
	return (1.0 + (rand / Math.sqrt((9.0 * (a + -0.3333333333333333))))) * (a + -0.3333333333333333);
}

Python

def code(a, rand):
	return (1.0 + (rand / math.sqrt((9.0 * (a + -0.3333333333333333))))) * (a + -0.3333333333333333)

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (1.0 + (rand / sqrt((9.0 * (a + -0.3333333333333333))))) * (a + -0.3333333333333333);
end

Wolfram

code[a_, rand_] := N[(N[(1.0 + N[(rand / N[Sqrt[N[(9.0 * N[(a + -0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + -0.3333333333333333), $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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -1.15 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ a -0.3333333333333333)) (* rand 0.3333333333333333))))
   (if (<= rand -1.15e+95)
     t_0
     (if (<= rand 1.35e+109) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -1.15e+95) {
		tmp = t_0;
	} else if (rand <= 1.35e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	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))) * (rand * 0.3333333333333333d0)
    if (rand <= (-1.15d+95)) then
        tmp = t_0
    else if (rand <= 1.35d+109) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -1.15e+95) {
		tmp = t_0;
	} else if (rand <= 1.35e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333)
	tmp = 0
	if rand <= -1.15e+95:
		tmp = t_0
	elif rand <= 1.35e+109:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(a + -0.3333333333333333)) * Float64(rand * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -1.15e+95)
		tmp = t_0;
	elseif (rand <= 1.35e+109)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -1.15e+95)
		tmp = t_0;
	elseif (rand <= 1.35e+109)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.15e+95], t$95$0, If[LessEqual[rand, 1.35e+109], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.14999999999999999e95 or 1.35000000000000001e109 < rand

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{a - \frac{1}{3}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a - \frac{1}{3}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \frac{-1}{3}\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{3} + a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      9. *-lowering-*.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right) \]

   7. Simplified 96.3%

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

   if -1.14999999999999999e95 < rand < 1.35000000000000001e109

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 93.5%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.15 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.95 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.4 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* 0.3333333333333333 (sqrt a)))))
   (if (<= rand -1.95e+95)
     t_0
     (if (<= rand 1.4e+109) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * sqrt(a));
	double tmp;
	if (rand <= -1.95e+95) {
		tmp = t_0;
	} else if (rand <= 1.4e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	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 = rand * (0.3333333333333333d0 * sqrt(a))
    if (rand <= (-1.95d+95)) then
        tmp = t_0
    else if (rand <= 1.4d+109) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * Math.sqrt(a));
	double tmp;
	if (rand <= -1.95e+95) {
		tmp = t_0;
	} else if (rand <= 1.4e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (0.3333333333333333 * math.sqrt(a))
	tmp = 0
	if rand <= -1.95e+95:
		tmp = t_0
	elif rand <= 1.4e+109:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(0.3333333333333333 * sqrt(a)))
	tmp = 0.0
	if (rand <= -1.95e+95)
		tmp = t_0;
	elseif (rand <= 1.4e+109)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (0.3333333333333333 * sqrt(a));
	tmp = 0.0;
	if (rand <= -1.95e+95)
		tmp = t_0;
	elseif (rand <= 1.4e+109)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.95e+95], t$95$0, If[LessEqual[rand, 1.4e+109], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.9499999999999999e95 or 1.4000000000000001e109 < rand

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{3} \cdot \sqrt{\frac{1}{a}}\right) \cdot \color{blue}{rand}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{1}{3}\right) \cdot rand\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right)\right) \]

      9. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right)\right) \]

   7. Simplified 97.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\sqrt{\frac{1}{a}} \cdot rand\right) \cdot \color{blue}{\frac{1}{3}}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(rand \cdot \frac{1}{3}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\sqrt{\frac{1}{a}} \cdot \left(\frac{1}{3} \cdot \color{blue}{rand}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right) \]

      7. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right) \]

   10. Simplified 94.6%

       \[\leadsto a \cdot \color{blue}{\left(\sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot \sqrt{\frac{1}{a}}\right) \cdot \frac{1}{3}\right), \color{blue}{rand}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{a}} \cdot a\right) \cdot \frac{1}{3}\right), rand\right) \]

       5. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\left(\frac{1}{a}\right)}^{\frac{1}{2}} \cdot a\right) \cdot \frac{1}{3}\right), rand\right) \]

       6. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\left({a}^{-1}\right)}^{\frac{1}{2}} \cdot a\right) \cdot \frac{1}{3}\right), rand\right) \]

       7. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({a}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot a\right) \cdot \frac{1}{3}\right), rand\right) \]

       8. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\left({a}^{\left(-1 \cdot \frac{1}{2} + 1\right)} \cdot \frac{1}{3}\right), rand\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({a}^{\left(\frac{-1}{2} + 1\right)} \cdot \frac{1}{3}\right), rand\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\left({a}^{\frac{1}{2}} \cdot \frac{1}{3}\right), rand\right) \]

       11. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{a} \cdot \frac{1}{3}\right), rand\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{a}\right), \frac{1}{3}\right), rand\right) \]

       13. sqrt-lowering-sqrt.f64 94.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(a\right), \frac{1}{3}\right), rand\right) \]

   12. Applied egg-rr 94.7%

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

   if -1.9499999999999999e95 < rand < 1.4000000000000001e109

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 93.5%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.95 \cdot 10^{+95}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.4 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{if}\;rand \leq -4.1 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (* rand 0.3333333333333333) (sqrt a))))
   (if (<= rand -4.1e+101)
     t_0
     (if (<= rand 1.35e+109) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = (rand * 0.3333333333333333) * sqrt(a);
	double tmp;
	if (rand <= -4.1e+101) {
		tmp = t_0;
	} else if (rand <= 1.35e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	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 = (rand * 0.3333333333333333d0) * sqrt(a)
    if (rand <= (-4.1d+101)) then
        tmp = t_0
    else if (rand <= 1.35d+109) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = (rand * 0.3333333333333333) * Math.sqrt(a);
	double tmp;
	if (rand <= -4.1e+101) {
		tmp = t_0;
	} else if (rand <= 1.35e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = (rand * 0.3333333333333333) * math.sqrt(a)
	tmp = 0
	if rand <= -4.1e+101:
		tmp = t_0
	elif rand <= 1.35e+109:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(Float64(rand * 0.3333333333333333) * sqrt(a))
	tmp = 0.0
	if (rand <= -4.1e+101)
		tmp = t_0;
	elseif (rand <= 1.35e+109)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = (rand * 0.3333333333333333) * sqrt(a);
	tmp = 0.0;
	if (rand <= -4.1e+101)
		tmp = t_0;
	elseif (rand <= 1.35e+109)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -4.1e+101], t$95$0, If[LessEqual[rand, 1.35e+109], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -4.1e101 or 1.35000000000000001e109 < rand

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{3} \cdot \sqrt{\frac{1}{a}}\right) \cdot \color{blue}{rand}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{1}{3}\right) \cdot rand\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right)\right) \]

      9. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right)\right) \]

   7. Simplified 97.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\sqrt{\frac{1}{a}} \cdot rand\right) \cdot \color{blue}{\frac{1}{3}}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(rand \cdot \frac{1}{3}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\sqrt{\frac{1}{a}} \cdot \left(\frac{1}{3} \cdot \color{blue}{rand}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right) \]

      7. *-lowering-*.f64 94.6%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right) \]

   10. Simplified 94.6%

       \[\leadsto a \cdot \color{blue}{\left(\sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. pow1/2 N/A

          \[\leadsto \left(\frac{1}{3} \cdot rand\right) \cdot \left({\left(\frac{1}{a}\right)}^{\frac{1}{2}} \cdot a\right) \]

       5. inv-pow N/A

          \[\leadsto \left(\frac{1}{3} \cdot rand\right) \cdot \left({\left({a}^{-1}\right)}^{\frac{1}{2}} \cdot a\right) \]

       6. pow-pow N/A

          \[\leadsto \left(\frac{1}{3} \cdot rand\right) \cdot \left({a}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot a\right) \]

       7. pow-plus N/A

          \[\leadsto \left(\frac{1}{3} \cdot rand\right) \cdot {a}^{\color{blue}{\left(-1 \cdot \frac{1}{2} + 1\right)}} \]

       8. metadata-eval N/A

          \[\leadsto \left(\frac{1}{3} \cdot rand\right) \cdot {a}^{\left(\frac{-1}{2} + 1\right)} \]

       9. metadata-eval N/A

          \[\leadsto \left(\frac{1}{3} \cdot rand\right) \cdot {a}^{\frac{1}{2}} \]

       10. pow1/2 N/A

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

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{3} \cdot rand\right), \color{blue}{\left(\sqrt{a}\right)}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\left(rand \cdot \frac{1}{3}\right), \left(\sqrt{\color{blue}{a}}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(rand, \frac{1}{3}\right), \left(\sqrt{\color{blue}{a}}\right)\right) \]

       14. sqrt-lowering-sqrt.f64 94.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(rand, \frac{1}{3}\right), \mathsf{sqrt.f64}\left(a\right)\right) \]

   12. Applied egg-rr 94.6%

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

   if -4.1e101 < rand < 1.35000000000000001e109

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 93.5%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.1 \cdot 10^{+101}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -1.4e+98)
     t_0
     (if (<= rand 1.35e+109) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -1.4e+98) {
		tmp = t_0;
	} else if (rand <= 1.35e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	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 = 0.3333333333333333d0 * (rand * sqrt(a))
    if (rand <= (-1.4d+98)) then
        tmp = t_0
    else if (rand <= 1.35d+109) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * Math.sqrt(a));
	double tmp;
	if (rand <= -1.4e+98) {
		tmp = t_0;
	} else if (rand <= 1.35e+109) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.3333333333333333 * (rand * math.sqrt(a))
	tmp = 0
	if rand <= -1.4e+98:
		tmp = t_0
	elif rand <= 1.35e+109:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.3333333333333333 * Float64(rand * sqrt(a)))
	tmp = 0.0
	if (rand <= -1.4e+98)
		tmp = t_0;
	elseif (rand <= 1.35e+109)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.3333333333333333 * (rand * sqrt(a));
	tmp = 0.0;
	if (rand <= -1.4e+98)
		tmp = t_0;
	elseif (rand <= 1.35e+109)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.4e+98], t$95$0, If[LessEqual[rand, 1.35e+109], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.4e98 or 1.35000000000000001e109 < rand

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{3} \cdot \sqrt{\frac{1}{a}}\right) \cdot \color{blue}{rand}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{1}{3}\right) \cdot rand\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right)\right) \]

      9. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right)\right) \]

   7. Simplified 97.9%

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

   8. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{a} \cdot rand\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(rand \cdot \color{blue}{\sqrt{a}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \color{blue}{\left(\sqrt{a}\right)}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 94.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \mathsf{sqrt.f64}\left(a\right)\right)\right) \]

   10. Simplified 94.5%

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

   if -1.4e98 < rand < 1.35000000000000001e109

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 93.5%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+109}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 700000000000:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;a + \frac{rand}{3} \cdot \sqrt{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= a 700000000000.0)
   (+ a -0.3333333333333333)
   (+ a (* (/ rand 3.0) (sqrt a)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= 700000000000.0) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = a + ((rand / 3.0) * sqrt(a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (a <= 700000000000.0d0) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = a + ((rand / 3.0d0) * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (a <= 700000000000.0) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = a + ((rand / 3.0) * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if a <= 700000000000.0:
		tmp = a + -0.3333333333333333
	else:
		tmp = a + ((rand / 3.0) * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= 700000000000.0)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(a + Float64(Float64(rand / 3.0) * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (a <= 700000000000.0)
		tmp = a + -0.3333333333333333;
	else
		tmp = a + ((rand / 3.0) * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[a, 700000000000.0], N[(a + -0.3333333333333333), $MachinePrecision], N[(a + N[(N[(rand / 3.0), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7e11

   1. Initial program 99.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 76.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 76.0%

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

   if 7e11 < a

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{3} \cdot \sqrt{\frac{1}{a}}\right) \cdot \color{blue}{rand}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{1}{3}\right) \cdot rand\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right)\right) \]

      9. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right)\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{a \cdot \left(1 + \sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\sqrt{\frac{1}{a}} \cdot \left(\frac{1}{3} \cdot rand\right)\right) \cdot a\right), \color{blue}{a}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{rand}{3} \cdot \sqrt{a} + a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 700000000000:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;a + \frac{rand}{3} \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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 + (-0.3333333333333333d0)) + (sqrt((a + (-0.3333333333333333d0))) * (rand * 0.3333333333333333d0))
end function

Java

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 + -0.3333333333333333) + (math.sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333))

Julia

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

MATLAB

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

Wolfram

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-neg-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot \sqrt{a - \frac{1}{3}}\right) \cdot rand\right)\right), \color{blue}{\left(a - \frac{1}{3}\right)}\right) \]

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 8: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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))))
end function

Java

public static double code(double a, double rand) {
	return (a + -0.3333333333333333) * (1.0 + (rand / Math.sqrt((a * 9.0))));
}

Python

def code(a, rand):
	return (a + -0.3333333333333333) * (1.0 + (rand / math.sqrt((a * 9.0))))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (a + -0.3333333333333333) * (1.0 + (rand / sqrt((a * 9.0))));
end

Wolfram

code[a_, rand_] := N[(N[(a + -0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(rand / N[Sqrt[N[(a * 9.0), $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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in a around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{a}, 9\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.0%

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

Alternative 9: 75.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{-0.3333333333333333 - a}\\ t_1 := a \cdot \left(a \cdot a\right)\\ t_2 := a \cdot t\_1\\ t_3 := 0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot t\_1\right)\\ \mathbf{if}\;rand \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\left(t\_3 \cdot \left(81 + a \cdot \left(a \cdot \left(-729 + t\_2 \cdot \left(59049 + \left(a \cdot a\right) \cdot -531441\right)\right)\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_3 \cdot \left(81 + \left(a \cdot a\right) \cdot \left(-729 + t\_2 \cdot 59049\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- -0.3333333333333333 a)))
        (t_1 (* a (* a a)))
        (t_2 (* a t_1))
        (t_3 (- 0.0013717421124828531 (* a (* (* a a) t_1)))))
   (if (<= rand -5e+134)
     (*
      (*
       t_3
       (+
        81.0
        (* a (* a (+ -729.0 (* t_2 (+ 59049.0 (* (* a a) -531441.0))))))))
      t_0)
     (if (<= rand 9.6e+141)
       (+ a -0.3333333333333333)
       (* t_0 (* t_3 (+ 81.0 (* (* a a) (+ -729.0 (* t_2 59049.0))))))))))

C

double code(double a, double rand) {
	double t_0 = 1.0 / (-0.3333333333333333 - a);
	double t_1 = a * (a * a);
	double t_2 = a * t_1;
	double t_3 = 0.0013717421124828531 - (a * ((a * a) * t_1));
	double tmp;
	if (rand <= -5e+134) {
		tmp = (t_3 * (81.0 + (a * (a * (-729.0 + (t_2 * (59049.0 + ((a * a) * -531441.0)))))))) * t_0;
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (t_3 * (81.0 + ((a * a) * (-729.0 + (t_2 * 59049.0)))));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 / ((-0.3333333333333333d0) - a)
    t_1 = a * (a * a)
    t_2 = a * t_1
    t_3 = 0.0013717421124828531d0 - (a * ((a * a) * t_1))
    if (rand <= (-5d+134)) then
        tmp = (t_3 * (81.0d0 + (a * (a * ((-729.0d0) + (t_2 * (59049.0d0 + ((a * a) * (-531441.0d0))))))))) * t_0
    else if (rand <= 9.6d+141) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * (t_3 * (81.0d0 + ((a * a) * ((-729.0d0) + (t_2 * 59049.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 1.0 / (-0.3333333333333333 - a);
	double t_1 = a * (a * a);
	double t_2 = a * t_1;
	double t_3 = 0.0013717421124828531 - (a * ((a * a) * t_1));
	double tmp;
	if (rand <= -5e+134) {
		tmp = (t_3 * (81.0 + (a * (a * (-729.0 + (t_2 * (59049.0 + ((a * a) * -531441.0)))))))) * t_0;
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (t_3 * (81.0 + ((a * a) * (-729.0 + (t_2 * 59049.0)))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 1.0 / (-0.3333333333333333 - a)
	t_1 = a * (a * a)
	t_2 = a * t_1
	t_3 = 0.0013717421124828531 - (a * ((a * a) * t_1))
	tmp = 0
	if rand <= -5e+134:
		tmp = (t_3 * (81.0 + (a * (a * (-729.0 + (t_2 * (59049.0 + ((a * a) * -531441.0)))))))) * t_0
	elif rand <= 9.6e+141:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (t_3 * (81.0 + ((a * a) * (-729.0 + (t_2 * 59049.0)))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(1.0 / Float64(-0.3333333333333333 - a))
	t_1 = Float64(a * Float64(a * a))
	t_2 = Float64(a * t_1)
	t_3 = Float64(0.0013717421124828531 - Float64(a * Float64(Float64(a * a) * t_1)))
	tmp = 0.0
	if (rand <= -5e+134)
		tmp = Float64(Float64(t_3 * Float64(81.0 + Float64(a * Float64(a * Float64(-729.0 + Float64(t_2 * Float64(59049.0 + Float64(Float64(a * a) * -531441.0)))))))) * t_0);
	elseif (rand <= 9.6e+141)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(t_3 * Float64(81.0 + Float64(Float64(a * a) * Float64(-729.0 + Float64(t_2 * 59049.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 1.0 / (-0.3333333333333333 - a);
	t_1 = a * (a * a);
	t_2 = a * t_1;
	t_3 = 0.0013717421124828531 - (a * ((a * a) * t_1));
	tmp = 0.0;
	if (rand <= -5e+134)
		tmp = (t_3 * (81.0 + (a * (a * (-729.0 + (t_2 * (59049.0 + ((a * a) * -531441.0)))))))) * t_0;
	elseif (rand <= 9.6e+141)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (t_3 * (81.0 + ((a * a) * (-729.0 + (t_2 * 59049.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(1.0 / N[(-0.3333333333333333 - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(0.0013717421124828531 - N[(a * N[(N[(a * a), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -5e+134], N[(N[(t$95$3 * N[(81.0 + N[(a * N[(a * N[(-729.0 + N[(t$95$2 * N[(59049.0 + N[(N[(a * a), $MachinePrecision] * -531441.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[rand, 9.6e+141], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(t$95$3 * N[(81.0 + N[(N[(a * a), $MachinePrecision] * N[(-729.0 + N[(t$95$2 * 59049.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -4.99999999999999981e134

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]
   10. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right) \cdot \frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{729} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27}\right), \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{9} \cdot \frac{1}{9}\right), \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 0.2%

       \[\leadsto \color{blue}{\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{1}{0.012345679012345678 + \left(a \cdot a\right) \cdot \left(0.1111111111111111 + a \cdot a\right)}\right)} \cdot \frac{1}{-0.3333333333333333 - a} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{\left(81 + {a}^{2} \cdot \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) - 729\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left({a}^{2} \cdot \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left(\left(a \cdot a\right) \cdot \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left(a \cdot \left(a \cdot \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) - 729\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(a, \left(a \cdot \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) - 729\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) - 729\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) + \left(\mathsf{neg}\left(729\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right) + -729\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(-729 + {a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-729, \left({a}^{4} \cdot \left(59049 + -531441 \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(\left({a}^{4}\right), \left(59049 + -531441 \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 60.6%

       \[\leadsto \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \color{blue}{\left(81 + a \cdot \left(a \cdot \left(-729 + \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(59049 + \left(a \cdot a\right) \cdot -531441\right)\right)\right)\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]

   if -4.99999999999999981e134 < rand < 9.59999999999999989e141

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 89.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 89.4%

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

   if 9.59999999999999989e141 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]
   10. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right) \cdot \frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{729} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27}\right), \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{9} \cdot \frac{1}{9}\right), \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 6.5%

       \[\leadsto \color{blue}{\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{1}{0.012345679012345678 + \left(a \cdot a\right) \cdot \left(0.1111111111111111 + a \cdot a\right)}\right)} \cdot \frac{1}{-0.3333333333333333 - a} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{\left(81 + {a}^{2} \cdot \left(59049 \cdot {a}^{4} - 729\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left({a}^{2} \cdot \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left({a}^{2}\right), \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(59049 \cdot {a}^{4} + \left(\mathsf{neg}\left(729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(59049 \cdot {a}^{4} + -729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(-729 + 59049 \cdot {a}^{4}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \left(59049 \cdot {a}^{4}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left({a}^{4}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left({a}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left({a}^{2} \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(\left(a \cdot a\right) \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(a \cdot \left(a \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(a \cdot {a}^{3}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \left({a}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       21. *-lowering-*.f64 48.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 48.9%

       \[\leadsto \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \color{blue}{\left(81 + \left(a \cdot a\right) \cdot \left(-729 + 59049 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(81 + a \cdot \left(a \cdot \left(-729 + \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(59049 + \left(a \cdot a\right) \cdot -531441\right)\right)\right)\right)\right) \cdot \frac{1}{-0.3333333333333333 - a}\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(81 + \left(a \cdot a\right) \cdot \left(-729 + \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot 59049\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot a\right)\\ t_1 := \frac{1}{-0.3333333333333333 - a}\\ t_2 := 0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot t\_0\right)\\ \mathbf{if}\;rand \leq -9.2 \cdot 10^{+134}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(81 + \left(a \cdot a\right) \cdot -729\right)\right)\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(81 + \left(a \cdot a\right) \cdot \left(-729 + \left(a \cdot t\_0\right) \cdot 59049\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* a (* a a)))
        (t_1 (/ 1.0 (- -0.3333333333333333 a)))
        (t_2 (- 0.0013717421124828531 (* a (* (* a a) t_0)))))
   (if (<= rand -9.2e+134)
     (* t_1 (* t_2 (+ 81.0 (* (* a a) -729.0))))
     (if (<= rand 9.6e+141)
       (+ a -0.3333333333333333)
       (*
        t_1
        (* t_2 (+ 81.0 (* (* a a) (+ -729.0 (* (* a t_0) 59049.0))))))))))

C

double code(double a, double rand) {
	double t_0 = a * (a * a);
	double t_1 = 1.0 / (-0.3333333333333333 - a);
	double t_2 = 0.0013717421124828531 - (a * ((a * a) * t_0));
	double tmp;
	if (rand <= -9.2e+134) {
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * -729.0)));
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * (-729.0 + ((a * t_0) * 59049.0)))));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = a * (a * a)
    t_1 = 1.0d0 / ((-0.3333333333333333d0) - a)
    t_2 = 0.0013717421124828531d0 - (a * ((a * a) * t_0))
    if (rand <= (-9.2d+134)) then
        tmp = t_1 * (t_2 * (81.0d0 + ((a * a) * (-729.0d0))))
    else if (rand <= 9.6d+141) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_1 * (t_2 * (81.0d0 + ((a * a) * ((-729.0d0) + ((a * t_0) * 59049.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = a * (a * a);
	double t_1 = 1.0 / (-0.3333333333333333 - a);
	double t_2 = 0.0013717421124828531 - (a * ((a * a) * t_0));
	double tmp;
	if (rand <= -9.2e+134) {
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * -729.0)));
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * (-729.0 + ((a * t_0) * 59049.0)))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = a * (a * a)
	t_1 = 1.0 / (-0.3333333333333333 - a)
	t_2 = 0.0013717421124828531 - (a * ((a * a) * t_0))
	tmp = 0
	if rand <= -9.2e+134:
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * -729.0)))
	elif rand <= 9.6e+141:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * (-729.0 + ((a * t_0) * 59049.0)))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(a * Float64(a * a))
	t_1 = Float64(1.0 / Float64(-0.3333333333333333 - a))
	t_2 = Float64(0.0013717421124828531 - Float64(a * Float64(Float64(a * a) * t_0)))
	tmp = 0.0
	if (rand <= -9.2e+134)
		tmp = Float64(t_1 * Float64(t_2 * Float64(81.0 + Float64(Float64(a * a) * -729.0))));
	elseif (rand <= 9.6e+141)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(81.0 + Float64(Float64(a * a) * Float64(-729.0 + Float64(Float64(a * t_0) * 59049.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = a * (a * a);
	t_1 = 1.0 / (-0.3333333333333333 - a);
	t_2 = 0.0013717421124828531 - (a * ((a * a) * t_0));
	tmp = 0.0;
	if (rand <= -9.2e+134)
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * -729.0)));
	elseif (rand <= 9.6e+141)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_1 * (t_2 * (81.0 + ((a * a) * (-729.0 + ((a * t_0) * 59049.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(-0.3333333333333333 - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.0013717421124828531 - N[(a * N[(N[(a * a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -9.2e+134], N[(t$95$1 * N[(t$95$2 * N[(81.0 + N[(N[(a * a), $MachinePrecision] * -729.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 9.6e+141], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(81.0 + N[(N[(a * a), $MachinePrecision] * N[(-729.0 + N[(N[(a * t$95$0), $MachinePrecision] * 59049.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -9.1999999999999992e134

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]
   10. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right) \cdot \frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{729} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27}\right), \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{9} \cdot \frac{1}{9}\right), \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 0.2%

       \[\leadsto \color{blue}{\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{1}{0.012345679012345678 + \left(a \cdot a\right) \cdot \left(0.1111111111111111 + a \cdot a\right)}\right)} \cdot \frac{1}{-0.3333333333333333 - a} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{\left(81 + -729 \cdot {a}^{2}\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left(-729 \cdot {a}^{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left({a}^{2} \cdot -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left({a}^{2}\right), -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left(a \cdot a\right), -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 58.3%

       \[\leadsto \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \color{blue}{\left(81 + \left(a \cdot a\right) \cdot -729\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]

   if -9.1999999999999992e134 < rand < 9.59999999999999989e141

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 89.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 89.4%

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

   if 9.59999999999999989e141 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]
   10. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right) \cdot \frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{729} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27}\right), \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{9} \cdot \frac{1}{9}\right), \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 6.5%

       \[\leadsto \color{blue}{\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{1}{0.012345679012345678 + \left(a \cdot a\right) \cdot \left(0.1111111111111111 + a \cdot a\right)}\right)} \cdot \frac{1}{-0.3333333333333333 - a} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{\left(81 + {a}^{2} \cdot \left(59049 \cdot {a}^{4} - 729\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left({a}^{2} \cdot \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left({a}^{2}\right), \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(59049 \cdot {a}^{4} - 729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(59049 \cdot {a}^{4} + \left(\mathsf{neg}\left(729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(59049 \cdot {a}^{4} + -729\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(-729 + 59049 \cdot {a}^{4}\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \left(59049 \cdot {a}^{4}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left({a}^{4}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left({a}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left({a}^{2} \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(\left(a \cdot a\right) \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(a \cdot \left(a \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \left(a \cdot {a}^{3}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \left({a}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({a}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       21. *-lowering-*.f64 48.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-729, \mathsf{*.f64}\left(59049, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 48.9%

       \[\leadsto \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \color{blue}{\left(81 + \left(a \cdot a\right) \cdot \left(-729 + 59049 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -9.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(81 + \left(a \cdot a\right) \cdot -729\right)\right)\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(81 + \left(a \cdot a\right) \cdot \left(-729 + \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot 59049\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{-0.3333333333333333 - a}\\ t_1 := 0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{if}\;rand \leq -8.2 \cdot 10^{+134}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(81 + \left(a \cdot a\right) \cdot -729\right)\right)\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot 81\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- -0.3333333333333333 a)))
        (t_1 (- 0.0013717421124828531 (* a (* (* a a) (* a (* a a)))))))
   (if (<= rand -8.2e+134)
     (* t_0 (* t_1 (+ 81.0 (* (* a a) -729.0))))
     (if (<= rand 9.6e+141) (+ a -0.3333333333333333) (* t_0 (* t_1 81.0))))))

C

double code(double a, double rand) {
	double t_0 = 1.0 / (-0.3333333333333333 - a);
	double t_1 = 0.0013717421124828531 - (a * ((a * a) * (a * (a * a))));
	double tmp;
	if (rand <= -8.2e+134) {
		tmp = t_0 * (t_1 * (81.0 + ((a * a) * -729.0)));
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (t_1 * 81.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / ((-0.3333333333333333d0) - a)
    t_1 = 0.0013717421124828531d0 - (a * ((a * a) * (a * (a * a))))
    if (rand <= (-8.2d+134)) then
        tmp = t_0 * (t_1 * (81.0d0 + ((a * a) * (-729.0d0))))
    else if (rand <= 9.6d+141) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * (t_1 * 81.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 1.0 / (-0.3333333333333333 - a);
	double t_1 = 0.0013717421124828531 - (a * ((a * a) * (a * (a * a))));
	double tmp;
	if (rand <= -8.2e+134) {
		tmp = t_0 * (t_1 * (81.0 + ((a * a) * -729.0)));
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (t_1 * 81.0);
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 1.0 / (-0.3333333333333333 - a)
	t_1 = 0.0013717421124828531 - (a * ((a * a) * (a * (a * a))))
	tmp = 0
	if rand <= -8.2e+134:
		tmp = t_0 * (t_1 * (81.0 + ((a * a) * -729.0)))
	elif rand <= 9.6e+141:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (t_1 * 81.0)
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(1.0 / Float64(-0.3333333333333333 - a))
	t_1 = Float64(0.0013717421124828531 - Float64(a * Float64(Float64(a * a) * Float64(a * Float64(a * a)))))
	tmp = 0.0
	if (rand <= -8.2e+134)
		tmp = Float64(t_0 * Float64(t_1 * Float64(81.0 + Float64(Float64(a * a) * -729.0))));
	elseif (rand <= 9.6e+141)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(t_1 * 81.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 1.0 / (-0.3333333333333333 - a);
	t_1 = 0.0013717421124828531 - (a * ((a * a) * (a * (a * a))));
	tmp = 0.0;
	if (rand <= -8.2e+134)
		tmp = t_0 * (t_1 * (81.0 + ((a * a) * -729.0)));
	elseif (rand <= 9.6e+141)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (t_1 * 81.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(1.0 / N[(-0.3333333333333333 - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.0013717421124828531 - N[(a * N[(N[(a * a), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.2e+134], N[(t$95$0 * N[(t$95$1 * N[(81.0 + N[(N[(a * a), $MachinePrecision] * -729.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 9.6e+141], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * 81.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -8.2000000000000007e134

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]
   10. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right) \cdot \frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{729} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27}\right), \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{9} \cdot \frac{1}{9}\right), \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 0.2%

       \[\leadsto \color{blue}{\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{1}{0.012345679012345678 + \left(a \cdot a\right) \cdot \left(0.1111111111111111 + a \cdot a\right)}\right)} \cdot \frac{1}{-0.3333333333333333 - a} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{\left(81 + -729 \cdot {a}^{2}\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left(-729 \cdot {a}^{2}\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \left({a}^{2} \cdot -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left({a}^{2}\right), -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\left(a \cdot a\right), -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{+.f64}\left(81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), -729\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 58.3%

       \[\leadsto \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \color{blue}{\left(81 + \left(a \cdot a\right) \cdot -729\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]

   if -8.2000000000000007e134 < rand < 9.59999999999999989e141

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 89.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 89.4%

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

   if 9.59999999999999989e141 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]
   10. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right) \cdot \frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{729} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27}\right), \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{9} \cdot \frac{1}{9}\right), \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 6.5%

       \[\leadsto \color{blue}{\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{1}{0.012345679012345678 + \left(a \cdot a\right) \cdot \left(0.1111111111111111 + a \cdot a\right)}\right)} \cdot \frac{1}{-0.3333333333333333 - a} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{81}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 46.4%

       \[\leadsto \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \color{blue}{81}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(81 + \left(a \cdot a\right) \cdot -729\right)\right)\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot 81\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -2.1 \cdot 10^{+134}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right)\right)\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot 81\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -2.1e+134)
   (*
    (- 0.1111111111111111 (* a a))
    (+ -3.0 (* a (+ 9.0 (* a (+ -27.0 (* a 81.0)))))))
   (if (<= rand 9.6e+141)
     (+ a -0.3333333333333333)
     (*
      (/ 1.0 (- -0.3333333333333333 a))
      (* (- 0.0013717421124828531 (* a (* (* a a) (* a (* a a))))) 81.0)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -2.1e+134) {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))));
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.0013717421124828531 - (a * ((a * a) * (a * (a * a))))) * 81.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-2.1d+134)) then
        tmp = (0.1111111111111111d0 - (a * a)) * ((-3.0d0) + (a * (9.0d0 + (a * ((-27.0d0) + (a * 81.0d0))))))
    else if (rand <= 9.6d+141) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (1.0d0 / ((-0.3333333333333333d0) - a)) * ((0.0013717421124828531d0 - (a * ((a * a) * (a * (a * a))))) * 81.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -2.1e+134) {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))));
	} else if (rand <= 9.6e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.0013717421124828531 - (a * ((a * a) * (a * (a * a))))) * 81.0);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -2.1e+134:
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))))
	elif rand <= 9.6e+141:
		tmp = a + -0.3333333333333333
	else:
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.0013717421124828531 - (a * ((a * a) * (a * (a * a))))) * 81.0)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -2.1e+134)
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * 81.0)))))));
	elseif (rand <= 9.6e+141)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 / Float64(-0.3333333333333333 - a)) * Float64(Float64(0.0013717421124828531 - Float64(a * Float64(Float64(a * a) * Float64(a * Float64(a * a))))) * 81.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -2.1e+134)
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))));
	elseif (rand <= 9.6e+141)
		tmp = a + -0.3333333333333333;
	else
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.0013717421124828531 - (a * ((a * a) * (a * (a * a))))) * 81.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -2.1e+134], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-3.0 + N[(a * N[(9.0 + N[(a * N[(-27.0 + N[(a * 81.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 9.6e+141], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(1.0 / N[(-0.3333333333333333 - a), $MachinePrecision]), $MachinePrecision] * N[(N[(0.0013717421124828531 - N[(a * N[(N[(a * a), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 81.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.1000000000000001e134

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) + -3\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(-3 + \color{blue}{a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)}\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \color{blue}{\left(9 + a \cdot \left(81 \cdot a - 27\right)\right)}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \color{blue}{\left(a \cdot \left(81 \cdot a - 27\right)\right)}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left(81 \cdot a - 27\right)}\right)\right)\right)\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + \color{blue}{\left(\mathsf{neg}\left(27\right)\right)}\right)\right)\right)\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + -27\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + \color{blue}{81 \cdot a}\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \color{blue}{\left(81 \cdot a\right)}\right)\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(a \cdot \color{blue}{81}\right)\right)\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 58.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \color{blue}{81}\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 58.2%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(-3 + a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right)\right)} \]

   if -2.1000000000000001e134 < rand < 9.59999999999999989e141

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 89.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 89.4%

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

   if 9.59999999999999989e141 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]
   10. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right) \cdot \frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\frac{1}{9}}^{3} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{729} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27} - {\left(a \cdot a\right)}^{3}\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{27} \cdot \frac{-1}{27}\right), \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left({\left(a \cdot a\right)}^{3}\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       9. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \left(a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \left(a \cdot a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \left(\frac{1}{\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} \cdot \frac{1}{9} + \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{9} \cdot \frac{1}{9}\right), \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \frac{1}{9} \cdot \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 6.5%

       \[\leadsto \color{blue}{\left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{1}{0.012345679012345678 + \left(a \cdot a\right) \cdot \left(0.1111111111111111 + a \cdot a\right)}\right)} \cdot \frac{1}{-0.3333333333333333 - a} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{81}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 46.4%

       \[\leadsto \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \color{blue}{81}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.1 \cdot 10^{+134}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right)\right)\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.0013717421124828531 - a \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot 81\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.1111111111111111 - a \cdot a\\ \mathbf{if}\;rand \leq -4.5 \cdot 10^{+133}:\\ \;\;\;\;t\_0 \cdot \left(-3 + a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right)\right)\\ \mathbf{elif}\;rand \leq 3.3 \cdot 10^{+150}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- 0.1111111111111111 (* a a))))
   (if (<= rand -4.5e+133)
     (* t_0 (+ -3.0 (* a (+ 9.0 (* a (+ -27.0 (* a 81.0)))))))
     (if (<= rand 3.3e+150)
       (+ a -0.3333333333333333)
       (* t_0 (+ -3.0 (* a (+ 9.0 (* a -27.0)))))))))

C

double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -4.5e+133) {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))));
	} else if (rand <= 3.3e+150) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	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 = 0.1111111111111111d0 - (a * a)
    if (rand <= (-4.5d+133)) then
        tmp = t_0 * ((-3.0d0) + (a * (9.0d0 + (a * ((-27.0d0) + (a * 81.0d0))))))
    else if (rand <= 3.3d+150) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -4.5e+133) {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))));
	} else if (rand <= 3.3e+150) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.1111111111111111 - (a * a)
	tmp = 0
	if rand <= -4.5e+133:
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))))
	elif rand <= 3.3e+150:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.1111111111111111 - Float64(a * a))
	tmp = 0.0
	if (rand <= -4.5e+133)
		tmp = Float64(t_0 * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * 81.0)))))));
	elseif (rand <= 3.3e+150)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.1111111111111111 - (a * a);
	tmp = 0.0;
	if (rand <= -4.5e+133)
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * (-27.0 + (a * 81.0))))));
	elseif (rand <= 3.3e+150)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -4.5e+133], N[(t$95$0 * N[(-3.0 + N[(a * N[(9.0 + N[(a * N[(-27.0 + N[(a * 81.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 3.3e+150], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -4.49999999999999985e133

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) + -3\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(-3 + \color{blue}{a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)}\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \color{blue}{\left(9 + a \cdot \left(81 \cdot a - 27\right)\right)}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \color{blue}{\left(a \cdot \left(81 \cdot a - 27\right)\right)}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left(81 \cdot a - 27\right)}\right)\right)\right)\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + \color{blue}{\left(\mathsf{neg}\left(27\right)\right)}\right)\right)\right)\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + -27\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + \color{blue}{81 \cdot a}\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \color{blue}{\left(81 \cdot a\right)}\right)\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(a \cdot \color{blue}{81}\right)\right)\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 58.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \color{blue}{81}\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 58.2%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(-3 + a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right)\right)} \]

   if -4.49999999999999985e133 < rand < 3.29999999999999981e150

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 88.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 88.5%

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

   if 3.29999999999999981e150 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.2%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 38.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 38.7%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + -3\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(-3 + \color{blue}{a \cdot \left(9 + -27 \cdot a\right)}\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \color{blue}{\left(9 + -27 \cdot a\right)}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \color{blue}{\left(-27 \cdot a\right)}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot \color{blue}{-27}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 43.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{-27}\right)\right)\right)\right)\right) \]

   12. Simplified 43.1%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.5 \cdot 10^{+133}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right)\right)\\ \mathbf{elif}\;rand \leq 3.3 \cdot 10^{+150}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.1111111111111111 - a \cdot a\\ \mathbf{if}\;rand \leq -5 \cdot 10^{+134}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 8 \cdot 10^{+151}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- 0.1111111111111111 (* a a))))
   (if (<= rand -5e+134)
     (* t_0 (+ (* a 9.0) -3.0))
     (if (<= rand 8e+151)
       (+ a -0.3333333333333333)
       (* t_0 (+ -3.0 (* a (+ 9.0 (* a -27.0)))))))))

C

double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -5e+134) {
		tmp = t_0 * ((a * 9.0) + -3.0);
	} else if (rand <= 8e+151) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	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 = 0.1111111111111111d0 - (a * a)
    if (rand <= (-5d+134)) then
        tmp = t_0 * ((a * 9.0d0) + (-3.0d0))
    else if (rand <= 8d+151) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -5e+134) {
		tmp = t_0 * ((a * 9.0) + -3.0);
	} else if (rand <= 8e+151) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.1111111111111111 - (a * a)
	tmp = 0
	if rand <= -5e+134:
		tmp = t_0 * ((a * 9.0) + -3.0)
	elif rand <= 8e+151:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.1111111111111111 - Float64(a * a))
	tmp = 0.0
	if (rand <= -5e+134)
		tmp = Float64(t_0 * Float64(Float64(a * 9.0) + -3.0));
	elseif (rand <= 8e+151)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.1111111111111111 - (a * a);
	tmp = 0.0;
	if (rand <= -5e+134)
		tmp = t_0 * ((a * 9.0) + -3.0);
	elseif (rand <= 8e+151)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -5e+134], N[(t$95$0 * N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 8e+151], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -4.99999999999999981e134

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(9 \cdot a - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(9 \cdot a + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(9 \cdot a + -3\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(-3 + \color{blue}{9 \cdot a}\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(9 \cdot a\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \left(a \cdot \color{blue}{9}\right)\right)\right) \]

       6. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \color{blue}{9}\right)\right)\right) \]

   12. Simplified 55.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(-3 + a \cdot 9\right)} \]

   if -4.99999999999999981e134 < rand < 8.00000000000000014e151

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 88.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 88.5%

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

   if 8.00000000000000014e151 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.2%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 38.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 38.7%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + -3\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(-3 + \color{blue}{a \cdot \left(9 + -27 \cdot a\right)}\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \color{blue}{\left(9 + -27 \cdot a\right)}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \color{blue}{\left(-27 \cdot a\right)}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot \color{blue}{-27}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 43.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{-27}\right)\right)\right)\right)\right) \]

   12. Simplified 43.1%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 8 \cdot 10^{+151}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -2.7 \cdot 10^{+134}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 1.95 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -2.7e+134)
   (* (- 0.1111111111111111 (* a a)) (+ (* a 9.0) -3.0))
   (if (<= rand 1.95e+148) (+ a -0.3333333333333333) (* 9.0 (* a (* a a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -2.7e+134) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0);
	} else if (rand <= 1.95e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 9.0 * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-2.7d+134)) then
        tmp = (0.1111111111111111d0 - (a * a)) * ((a * 9.0d0) + (-3.0d0))
    else if (rand <= 1.95d+148) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 9.0d0 * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -2.7e+134) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0);
	} else if (rand <= 1.95e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 9.0 * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -2.7e+134:
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0)
	elif rand <= 1.95e+148:
		tmp = a + -0.3333333333333333
	else:
		tmp = 9.0 * (a * (a * a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -2.7e+134)
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(Float64(a * 9.0) + -3.0));
	elseif (rand <= 1.95e+148)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(9.0 * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -2.7e+134)
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0);
	elseif (rand <= 1.95e+148)
		tmp = a + -0.3333333333333333;
	else
		tmp = 9.0 * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -2.7e+134], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.95e+148], N[(a + -0.3333333333333333), $MachinePrecision], N[(9.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.7e134

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(9 \cdot a - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(9 \cdot a + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(9 \cdot a + -3\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(-3 + \color{blue}{9 \cdot a}\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(9 \cdot a\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \left(a \cdot \color{blue}{9}\right)\right)\right) \]

       6. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, \color{blue}{9}\right)\right)\right) \]

   12. Simplified 55.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(-3 + a \cdot 9\right)} \]

   if -2.7e134 < rand < 1.95000000000000001e148

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 88.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 88.5%

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

   if 1.95000000000000001e148 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.2%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{-1}{3} + \left(a \cdot a - \frac{-1}{3} \cdot a\right)}{{\frac{-1}{3}}^{3} + {a}^{3}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} + \left(a \cdot a - \frac{-1}{3} \cdot a\right)\right), \color{blue}{\left({\frac{-1}{3}}^{3} + {a}^{3}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a - \frac{-1}{3} \cdot a\right)\right), \left(\color{blue}{{\frac{-1}{3}}^{3}} + {a}^{3}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(a \cdot a - \frac{-1}{3} \cdot a\right)\right), \left({\color{blue}{\frac{-1}{3}}}^{3} + {a}^{3}\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(a \cdot \left(a - \frac{-1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{\color{blue}{3}} + {a}^{3}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \left(a - \frac{-1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{\color{blue}{3}} + {a}^{3}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \left(a + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right)\right), \left({\frac{-1}{3}}^{3} + {a}^{3}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \left(a + \frac{1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{3} + {a}^{3}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{3} + {a}^{3}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{-1}{3}}^{3}\right), \color{blue}{\left({a}^{3}\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \left({\color{blue}{a}}^{3}\right)\right)\right)\right) \]

      14. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}{-0.037037037037037035 + a \cdot \left(a \cdot a\right)}}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\frac{1}{9}}, \mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 40.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.1111111111111111}}{-0.037037037037037035 + a \cdot \left(a \cdot a\right)}} \]

   13. Taylor expanded in a around inf

       \[\leadsto \color{blue}{9 \cdot {a}^{3}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {a}^{3} \cdot \color{blue}{9} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({a}^{3}\right), \color{blue}{9}\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), 9\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot {a}^{2}\right), 9\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{2}\right)\right), 9\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), 9\right) \]

       7. *-lowering-*.f64 40.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), 9\right) \]

   15. Simplified 40.4%

       \[\leadsto \color{blue}{\left(a \cdot \left(a \cdot a\right)\right) \cdot 9} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.7 \cdot 10^{+134}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 1.95 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 4.3 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 4.3e+148) (+ a -0.3333333333333333) (* 9.0 (* a (* a a)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 4.3e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 9.0 * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 4.3d+148) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 9.0d0 * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 4.3e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 9.0 * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 4.3e+148:
		tmp = a + -0.3333333333333333
	else:
		tmp = 9.0 * (a * (a * a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 4.3e+148)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(9.0 * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 4.3e+148)
		tmp = a + -0.3333333333333333;
	else
		tmp = 9.0 * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 4.3e+148], N[(a + -0.3333333333333333), $MachinePrecision], N[(9.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 4.3000000000000002e148

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 72.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 72.6%

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

   if 4.3000000000000002e148 < rand

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.2%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{-1}{3} + \left(a \cdot a - \frac{-1}{3} \cdot a\right)}{{\frac{-1}{3}}^{3} + {a}^{3}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} + \left(a \cdot a - \frac{-1}{3} \cdot a\right)\right), \color{blue}{\left({\frac{-1}{3}}^{3} + {a}^{3}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a - \frac{-1}{3} \cdot a\right)\right), \left(\color{blue}{{\frac{-1}{3}}^{3}} + {a}^{3}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(a \cdot a - \frac{-1}{3} \cdot a\right)\right), \left({\color{blue}{\frac{-1}{3}}}^{3} + {a}^{3}\right)\right)\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(a \cdot \left(a - \frac{-1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{\color{blue}{3}} + {a}^{3}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \left(a - \frac{-1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{\color{blue}{3}} + {a}^{3}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \left(a + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right)\right), \left({\frac{-1}{3}}^{3} + {a}^{3}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \left(a + \frac{1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{3} + {a}^{3}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \left({\frac{-1}{3}}^{3} + {a}^{3}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\left({\frac{-1}{3}}^{3}\right), \color{blue}{\left({a}^{3}\right)}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \left({\color{blue}{a}}^{3}\right)\right)\right)\right) \]

      14. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}{-0.037037037037037035 + a \cdot \left(a \cdot a\right)}}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{\frac{1}{9}}, \mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 40.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{0.1111111111111111}}{-0.037037037037037035 + a \cdot \left(a \cdot a\right)}} \]

   13. Taylor expanded in a around inf

       \[\leadsto \color{blue}{9 \cdot {a}^{3}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {a}^{3} \cdot \color{blue}{9} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({a}^{3}\right), \color{blue}{9}\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(a \cdot a\right)\right), 9\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(a \cdot {a}^{2}\right), 9\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({a}^{2}\right)\right), 9\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot a\right)\right), 9\right) \]

       7. *-lowering-*.f64 40.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right), 9\right) \]

   15. Simplified 40.4%

       \[\leadsto \color{blue}{\left(a \cdot \left(a \cdot a\right)\right) \cdot 9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 4.3 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.2% accurate, 39.7× speedup

Math

\[\begin{array}{l} \\ a + -0.3333333333333333 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return a + -0.3333333333333333

Julia

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

MATLAB

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

Wolfram

code[a_, rand_] := N[(a + -0.3333333333333333), $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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 63.7%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 63.7%

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

8. Final simplification 63.7%

   \[\leadsto a + -0.3333333333333333 \]
9. Add Preprocessing

Alternative 18: 62.2% accurate, 119.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 a)

C

double code(double a, double rand) {
	return a;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a
end function

Java

public static double code(double a, double rand) {
	return a;
}

Python

def code(a, rand):
	return a

Julia

function code(a, rand)
	return a
end

MATLAB

function tmp = code(a, rand)
	tmp = a;
end

Wolfram

code[a_, rand_] := a

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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 63.7%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 63.7%

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

8. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 61.7%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Alternative 19: 1.5% accurate, 119.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 -0.3333333333333333)

C

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

Fortran

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

Java

public static double code(double a, double rand) {
	return -0.3333333333333333;
}

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 63.7%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 63.7%

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

8. Taylor expanded in a around 0

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

10. Simplified 1.5%

    \[\leadsto \color{blue}{-0.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))))