Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 12.0s

Alternatives: 22

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

\[\begin{array}{l} \\ \left(-0.3333333333333333 + a\right) + \frac{{\left(-0.3333333333333333 + a\right)}^{0.5}}{\frac{3}{rand}} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (+
  (+ -0.3333333333333333 a)
  (/ (pow (+ -0.3333333333333333 a) 0.5) (/ 3.0 rand))))

C

double code(double a, double rand) {
	return (-0.3333333333333333 + a) + (pow((-0.3333333333333333 + a), 0.5) / (3.0 / rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = ((-0.3333333333333333d0) + a) + ((((-0.3333333333333333d0) + a) ** 0.5d0) / (3.0d0 / rand))
end function

Java

public static double code(double a, double rand) {
	return (-0.3333333333333333 + a) + (Math.pow((-0.3333333333333333 + a), 0.5) / (3.0 / rand));
}

Python

def code(a, rand):
	return (-0.3333333333333333 + a) + (math.pow((-0.3333333333333333 + a), 0.5) / (3.0 / rand))

Julia

function code(a, rand)
	return Float64(Float64(-0.3333333333333333 + a) + Float64((Float64(-0.3333333333333333 + a) ^ 0.5) / Float64(3.0 / rand)))
end

MATLAB

function tmp = code(a, rand)
	tmp = (-0.3333333333333333 + a) + (((-0.3333333333333333 + a) ^ 0.5) / (3.0 / rand));
end

Wolfram

code[a_, rand_] := N[(N[(-0.3333333333333333 + a), $MachinePrecision] + N[(N[Power[N[(-0.3333333333333333 + a), $MachinePrecision], 0.5], $MachinePrecision] / N[(3.0 / rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

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

   1. +-commutative N/A

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

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

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

   3. pow1/2 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

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

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

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

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

   10. +-commutative N/A

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

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

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

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

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

   13. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right), \frac{1}{2}\right), \mathsf{/.f64}\left(3, rand\right)\right), \left(\frac{-1}{3} + \color{blue}{a}\right)\right) \]

   14. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right), \frac{1}{2}\right), \mathsf{/.f64}\left(3, rand\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right) \]

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{{\left(-0.3333333333333333 + a\right)}^{0.5}}{\frac{3}{rand}} + \left(-0.3333333333333333 + a\right)} \]

10. Final simplification 99.9%

    \[\leadsto \left(-0.3333333333333333 + a\right) + \frac{{\left(-0.3333333333333333 + a\right)}^{0.5}}{\frac{3}{rand}} \]
11. Add Preprocessing

Alternative 2: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-0.3333333333333333 + a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -3.2 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 6 \cdot 10^{+85}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ -0.3333333333333333 a)) (* rand 0.3333333333333333))))
   (if (<= rand -3.2e+88)
     t_0
     (if (<= rand 6e+85) (+ -0.3333333333333333 a) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((-0.3333333333333333 + a)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -3.2e+88) {
		tmp = t_0;
	} else if (rand <= 6e+85) {
		tmp = -0.3333333333333333 + a;
	} 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(((-0.3333333333333333d0) + a)) * (rand * 0.3333333333333333d0)
    if (rand <= (-3.2d+88)) then
        tmp = t_0
    else if (rand <= 6d+85) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((-0.3333333333333333 + a)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -3.2e+88) {
		tmp = t_0;
	} else if (rand <= 6e+85) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((-0.3333333333333333 + a)) * (rand * 0.3333333333333333)
	tmp = 0
	if rand <= -3.2e+88:
		tmp = t_0
	elif rand <= 6e+85:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(-0.3333333333333333 + a)) * Float64(rand * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -3.2e+88)
		tmp = t_0;
	elseif (rand <= 6e+85)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((-0.3333333333333333 + a)) * (rand * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -3.2e+88)
		tmp = t_0;
	elseif (rand <= 6e+85)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[N[(-0.3333333333333333 + a), $MachinePrecision]], $MachinePrecision] * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -3.2e+88], t$95$0, If[LessEqual[rand, 6e+85], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -3.1999999999999999e88 or 6.0000000000000001e85 < 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 94.6%

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

   7. Simplified 94.6%

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

   if -3.1999999999999999e88 < rand < 6.0000000000000001e85

   1. Initial program 100.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 93.1%

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

   7. Simplified 93.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -3.2 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{-0.3333333333333333 + a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 6 \cdot 10^{+85}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.3333333333333333 + a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -9.5 \cdot 10^{+90}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 1.32 \cdot 10^{+87}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -9.5e+90)
   (* (* rand 0.3333333333333333) (sqrt a))
   (if (<= rand 1.32e+87)
     (+ -0.3333333333333333 a)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -9.5e+90) {
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	} else if (rand <= 1.32e+87) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 0.3333333333333333 * (rand * 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 (rand <= (-9.5d+90)) then
        tmp = (rand * 0.3333333333333333d0) * sqrt(a)
    else if (rand <= 1.32d+87) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -9.5e+90) {
		tmp = (rand * 0.3333333333333333) * Math.sqrt(a);
	} else if (rand <= 1.32e+87) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -9.5e+90:
		tmp = (rand * 0.3333333333333333) * math.sqrt(a)
	elif rand <= 1.32e+87:
		tmp = -0.3333333333333333 + a
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -9.5e+90)
		tmp = Float64(Float64(rand * 0.3333333333333333) * sqrt(a));
	elseif (rand <= 1.32e+87)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -9.5e+90)
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	elseif (rand <= 1.32e+87)
		tmp = -0.3333333333333333 + a;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -9.5e+90], N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.32e+87], N[(-0.3333333333333333 + a), $MachinePrecision], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -9.4999999999999994e90

   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 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 94.1%

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

   7. Simplified 94.1%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 93.9%

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

   if -9.4999999999999994e90 < rand < 1.32e87

   1. Initial program 100.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 93.1%

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

   7. Simplified 93.1%

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

   if 1.32e87 < 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 99.7%

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

   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 99.1%

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

   7. Simplified 99.1%

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

   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 94.7%

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

   10. Simplified 94.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -9.5 \cdot 10^{+90}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 1.32 \cdot 10^{+87}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.2 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 7.8 \cdot 10^{+88}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -1.2e+89)
     t_0
     (if (<= rand 7.8e+88) (+ -0.3333333333333333 a) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -1.2e+89) {
		tmp = t_0;
	} else if (rand <= 7.8e+88) {
		tmp = -0.3333333333333333 + a;
	} 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 <= (-1.2d+89)) then
        tmp = t_0
    else if (rand <= 7.8d+88) then
        tmp = (-0.3333333333333333d0) + a
    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 <= -1.2e+89) {
		tmp = t_0;
	} else if (rand <= 7.8e+88) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.3333333333333333 * (rand * math.sqrt(a))
	tmp = 0
	if rand <= -1.2e+89:
		tmp = t_0
	elif rand <= 7.8e+88:
		tmp = -0.3333333333333333 + a
	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 <= -1.2e+89)
		tmp = t_0;
	elseif (rand <= 7.8e+88)
		tmp = Float64(-0.3333333333333333 + a);
	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 <= -1.2e+89)
		tmp = t_0;
	elseif (rand <= 7.8e+88)
		tmp = -0.3333333333333333 + a;
	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, -1.2e+89], t$95$0, If[LessEqual[rand, 7.8e+88], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.20000000000000002e89 or 7.8000000000000002e88 < 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 99.2%

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

   7. Simplified 99.2%

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

   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 94.3%

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

   10. Simplified 94.3%

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

   if -1.20000000000000002e89 < rand < 7.8000000000000002e88

   1. Initial program 100.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 93.1%

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

   7. Simplified 93.1%

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

Alternative 5: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = ((-0.3333333333333333d0) + a) + (sqrt(((-0.3333333333333333d0) + a)) * (rand / 3.0d0))
end function

Java

public static double code(double a, double rand) {
	return (-0.3333333333333333 + a) + (Math.sqrt((-0.3333333333333333 + a)) * (rand / 3.0));
}

Python

def code(a, rand):
	return (-0.3333333333333333 + a) + (math.sqrt((-0.3333333333333333 + a)) * (rand / 3.0))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (-0.3333333333333333 + a) + (sqrt((-0.3333333333333333 + a)) * (rand / 3.0));
end

Wolfram

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

Derivation

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

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. /-lowering-/.f64 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \left(-0.3333333333333333 + a\right) + \sqrt{-0.3333333333333333 + a} \cdot \frac{rand}{3} \]
11. Add Preprocessing

Alternative 6: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

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

Alternative 7: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ a + \frac{{\left(-0.3333333333333333 + a\right)}^{0.5}}{\frac{3}{rand}} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (+ a (/ (pow (+ -0.3333333333333333 a) 0.5) (/ 3.0 rand))))

C

double code(double a, double rand) {
	return a + (pow((-0.3333333333333333 + a), 0.5) / (3.0 / rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a + ((((-0.3333333333333333d0) + a) ** 0.5d0) / (3.0d0 / rand))
end function

Java

public static double code(double a, double rand) {
	return a + (Math.pow((-0.3333333333333333 + a), 0.5) / (3.0 / rand));
}

Python

def code(a, rand):
	return a + (math.pow((-0.3333333333333333 + a), 0.5) / (3.0 / rand))

Julia

function code(a, rand)
	return Float64(a + Float64((Float64(-0.3333333333333333 + a) ^ 0.5) / Float64(3.0 / rand)))
end

MATLAB

function tmp = code(a, rand)
	tmp = a + (((-0.3333333333333333 + a) ^ 0.5) / (3.0 / rand));
end

Wolfram

code[a_, rand_] := N[(a + N[(N[Power[N[(-0.3333333333333333 + a), $MachinePrecision], 0.5], $MachinePrecision] / N[(3.0 / rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

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

   1. +-commutative N/A

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

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

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

   3. pow1/2 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. un-div-inv N/A

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

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

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

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

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

   10. +-commutative N/A

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

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

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

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

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

   13. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right), \frac{1}{2}\right), \mathsf{/.f64}\left(3, rand\right)\right), \left(\frac{-1}{3} + \color{blue}{a}\right)\right) \]

   14. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right), \frac{1}{2}\right), \mathsf{/.f64}\left(3, rand\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right) \]

9. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{{\left(-0.3333333333333333 + a\right)}^{0.5}}{\frac{3}{rand}} + \left(-0.3333333333333333 + a\right)} \]

10. Taylor expanded in a around inf

    \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right), \frac{1}{2}\right), \mathsf{/.f64}\left(3, rand\right)\right), \color{blue}{a}\right) \]
11. Step-by-step derivation

12. Simplified 99.5%

    \[\leadsto \frac{{\left(-0.3333333333333333 + a\right)}^{0.5}}{\frac{3}{rand}} + \color{blue}{a} \]

13. Final simplification 99.5%

    \[\leadsto a + \frac{{\left(-0.3333333333333333 + a\right)}^{0.5}}{\frac{3}{rand}} \]
14. Add Preprocessing

Alternative 8: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ a + \frac{\sqrt{a}}{\frac{3}{rand}} \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 (+ a (/ (sqrt a) (/ 3.0 rand))))

C

double code(double a, double rand) {
	return a + (sqrt(a) / (3.0 / rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a + (sqrt(a) / (3.0d0 / rand))
end function

Java

public static double code(double a, double rand) {
	return a + (Math.sqrt(a) / (3.0 / rand));
}

Python

def code(a, rand):
	return a + (math.sqrt(a) / (3.0 / rand))

Julia

function code(a, rand)
	return Float64(a + Float64(sqrt(a) / Float64(3.0 / rand)))
end

MATLAB

function tmp = code(a, rand)
	tmp = a + (sqrt(a) / (3.0 / rand));
end

Wolfram

code[a_, rand_] := N[(a + N[(N[Sqrt[a], $MachinePrecision] / N[(3.0 / rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 99.0%

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

7. Simplified 99.0%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. pow1/2 N/A

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

   11. inv-pow N/A

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

   12. pow-pow N/A

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

   13. pow-plus N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. pow1/2 N/A

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

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

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

   18. /-lowering-/.f64 N/A

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

   19. sqrt-lowering-sqrt.f64 99.1%

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

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{rand}{3} \cdot \sqrt{a} + a} \]
10. Step-by-step derivation

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

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

    2. *-commutative N/A

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

    3. clear-num N/A

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

    4. un-div-inv N/A

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

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

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

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

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

    7. /-lowering-/.f64 99.1%

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

11. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{\frac{\sqrt{a}}{\frac{3}{rand}} + a} \]

12. Final simplification 99.1%

    \[\leadsto a + \frac{\sqrt{a}}{\frac{3}{rand}} \]
13. Add Preprocessing

Alternative 9: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ a + rand \cdot \frac{\sqrt{a}}{3} \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 (+ a (* rand (/ (sqrt a) 3.0))))

C

double code(double a, double rand) {
	return a + (rand * (sqrt(a) / 3.0));
}

Fortran

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

Java

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

Python

def code(a, rand):
	return a + (rand * (math.sqrt(a) / 3.0))

Julia

function code(a, rand)
	return Float64(a + Float64(rand * Float64(sqrt(a) / 3.0)))
end

MATLAB

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

Wolfram

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

Derivation

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 99.0%

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

7. Simplified 99.0%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. pow1/2 N/A

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

   11. inv-pow N/A

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

   12. pow-pow N/A

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

   13. pow-plus N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. pow1/2 N/A

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

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

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

   18. /-lowering-/.f64 N/A

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

   19. sqrt-lowering-sqrt.f64 99.1%

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

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{rand}{3} \cdot \sqrt{a} + a} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

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

    2. associate-/l* N/A

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

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

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

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

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

    5. sqrt-lowering-sqrt.f64 99.1%

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

11. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{rand \cdot \frac{\sqrt{a}}{3}} + a \]

12. Final simplification 99.1%

    \[\leadsto a + rand \cdot \frac{\sqrt{a}}{3} \]
13. Add Preprocessing

Alternative 10: 75.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{-0.3333333333333333 - a}\\ t_1 := 0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;\left(t\_1 \cdot \left(9 + \left(a \cdot a\right) \cdot \left(-81 + \left(a \cdot a\right) \cdot \left(729 + \left(a \cdot a\right) \cdot -6561\right)\right)\right)\right) \cdot t\_0\\ \mathbf{elif}\;rand \leq 2 \cdot 10^{+154}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(9 + a \cdot \left(a \cdot \left(-81 + \left(a \cdot a\right) \cdot 729\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- -0.3333333333333333 a)))
        (t_1 (- 0.012345679012345678 (* (* a a) (* a a)))))
   (if (<= rand -2.25e+126)
     (*
      (*
       t_1
       (+ 9.0 (* (* a a) (+ -81.0 (* (* a a) (+ 729.0 (* (* a a) -6561.0)))))))
      t_0)
     (if (<= rand 2e+154)
       (+ -0.3333333333333333 a)
       (* t_0 (* t_1 (+ 9.0 (* a (* a (+ -81.0 (* (* a a) 729.0)))))))))))

C

double code(double a, double rand) {
	double t_0 = 1.0 / (-0.3333333333333333 - a);
	double t_1 = 0.012345679012345678 - ((a * a) * (a * a));
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = (t_1 * (9.0 + ((a * a) * (-81.0 + ((a * a) * (729.0 + ((a * a) * -6561.0))))))) * t_0;
	} else if (rand <= 2e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (t_1 * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))));
	}
	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 = 0.012345679012345678d0 - ((a * a) * (a * a))
    if (rand <= (-2.25d+126)) then
        tmp = (t_1 * (9.0d0 + ((a * a) * ((-81.0d0) + ((a * a) * (729.0d0 + ((a * a) * (-6561.0d0)))))))) * t_0
    else if (rand <= 2d+154) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0 * (t_1 * (9.0d0 + (a * (a * ((-81.0d0) + ((a * a) * 729.0d0))))))
    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 = 0.012345679012345678 - ((a * a) * (a * a));
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = (t_1 * (9.0 + ((a * a) * (-81.0 + ((a * a) * (729.0 + ((a * a) * -6561.0))))))) * t_0;
	} else if (rand <= 2e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (t_1 * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 1.0 / (-0.3333333333333333 - a)
	t_1 = 0.012345679012345678 - ((a * a) * (a * a))
	tmp = 0
	if rand <= -2.25e+126:
		tmp = (t_1 * (9.0 + ((a * a) * (-81.0 + ((a * a) * (729.0 + ((a * a) * -6561.0))))))) * t_0
	elif rand <= 2e+154:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0 * (t_1 * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(1.0 / Float64(-0.3333333333333333 - a))
	t_1 = Float64(0.012345679012345678 - Float64(Float64(a * a) * Float64(a * a)))
	tmp = 0.0
	if (rand <= -2.25e+126)
		tmp = Float64(Float64(t_1 * Float64(9.0 + Float64(Float64(a * a) * Float64(-81.0 + Float64(Float64(a * a) * Float64(729.0 + Float64(Float64(a * a) * -6561.0))))))) * t_0);
	elseif (rand <= 2e+154)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(9.0 + Float64(a * Float64(a * Float64(-81.0 + Float64(Float64(a * a) * 729.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 1.0 / (-0.3333333333333333 - a);
	t_1 = 0.012345679012345678 - ((a * a) * (a * a));
	tmp = 0.0;
	if (rand <= -2.25e+126)
		tmp = (t_1 * (9.0 + ((a * a) * (-81.0 + ((a * a) * (729.0 + ((a * a) * -6561.0))))))) * t_0;
	elseif (rand <= 2e+154)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0 * (t_1 * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))));
	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[(0.012345679012345678 - N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -2.25e+126], N[(N[(t$95$1 * N[(9.0 + N[(N[(a * a), $MachinePrecision] * N[(-81.0 + N[(N[(a * a), $MachinePrecision] * N[(729.0 + N[(N[(a * a), $MachinePrecision] * -6561.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[rand, 2e+154], N[(-0.3333333333333333 + a), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(9.0 + N[(a * N[(a * N[(-81.0 + N[(N[(a * a), $MachinePrecision] * 729.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.24999999999999987e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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 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. flip-- N/A

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

       2. div-inv N/A

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

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

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

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

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} + a \cdot a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 0.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 0.1%

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

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + {a}^{2} \cdot \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right) - 81\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left({a}^{2} \cdot \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right) - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\left({a}^{2}\right), \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right) - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\left(a \cdot a\right), \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right) - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right) - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right) + \left(\mathsf{neg}\left(81\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right) + -81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(-81 + {a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \left({a}^{2} \cdot \left(729 + -6561 \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\left({a}^{2}\right), \left(729 + -6561 \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(729 + -6561 \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(729 + -6561 \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(729, \left(-6561 \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(729, \left({a}^{2} \cdot -6561\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(729, \mathsf{*.f64}\left(\left({a}^{2}\right), -6561\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(729, \mathsf{*.f64}\left(\left(a \cdot a\right), -6561\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       16. *-lowering-*.f64 49.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(729, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), -6561\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 49.9%

       \[\leadsto \left(\left(0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + \left(a \cdot a\right) \cdot \left(-81 + \left(a \cdot a\right) \cdot \left(729 + \left(a \cdot a\right) \cdot -6561\right)\right)\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]

   if -2.24999999999999987e126 < rand < 2.00000000000000007e154

   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 84.2%

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

   7. Simplified 84.2%

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

   if 2.00000000000000007e154 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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 44.5%

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

   9. Applied egg-rr 44.5%

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

       1. flip-- N/A

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

       2. div-inv N/A

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

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

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

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

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} + a \cdot a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 19.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 19.8%

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

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + {a}^{2} \cdot \left(729 \cdot {a}^{2} - 81\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left({a}^{2} \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left(\left(a \cdot a\right) \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left(a \cdot \left(a \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(a \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(729 \cdot {a}^{2} + \left(\mathsf{neg}\left(81\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(729 \cdot {a}^{2} + -81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(-81 + 729 \cdot {a}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \left(729 \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \left({a}^{2} \cdot 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\left({a}^{2}\right), 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\left(a \cdot a\right), 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 62.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 62.8%

       \[\leadsto \left(\left(0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot \left(a \cdot \left(-81 + \left(a \cdot a\right) \cdot 729\right)\right)\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;\left(\left(0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(9 + \left(a \cdot a\right) \cdot \left(-81 + \left(a \cdot a\right) \cdot \left(729 + \left(a \cdot a\right) \cdot -6561\right)\right)\right)\right) \cdot \frac{1}{-0.3333333333333333 - a}\\ \mathbf{elif}\;rand \leq 2 \cdot 10^{+154}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot \left(a \cdot \left(-81 + \left(a \cdot a\right) \cdot 729\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.85 \cdot 10^{+126}:\\ \;\;\;\;\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot \left(-27 + \left(a \cdot a\right) \cdot \left(243 + a \cdot -729\right)\right)\right)\\ \mathbf{elif}\;rand \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot \left(a \cdot \left(-81 + \left(a \cdot a\right) \cdot 729\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.85e+126)
   (*
    (+ -0.037037037037037035 (* a (* a a)))
    (+ 9.0 (* a (+ -27.0 (* (* a a) (+ 243.0 (* a -729.0)))))))
   (if (<= rand 1.4e+154)
     (+ -0.3333333333333333 a)
     (*
      (/ 1.0 (- -0.3333333333333333 a))
      (*
       (- 0.012345679012345678 (* (* a a) (* a a)))
       (+ 9.0 (* a (* a (+ -81.0 (* (* a a) 729.0))))))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.85e+126) {
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))));
	} else if (rand <= 1.4e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.012345679012345678 - ((a * a) * (a * a))) * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.85d+126)) then
        tmp = ((-0.037037037037037035d0) + (a * (a * a))) * (9.0d0 + (a * ((-27.0d0) + ((a * a) * (243.0d0 + (a * (-729.0d0)))))))
    else if (rand <= 1.4d+154) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (1.0d0 / ((-0.3333333333333333d0) - a)) * ((0.012345679012345678d0 - ((a * a) * (a * a))) * (9.0d0 + (a * (a * ((-81.0d0) + ((a * a) * 729.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.85e+126) {
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))));
	} else if (rand <= 1.4e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.012345679012345678 - ((a * a) * (a * a))) * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.85e+126:
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))))
	elif rand <= 1.4e+154:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.012345679012345678 - ((a * a) * (a * a))) * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.85e+126)
		tmp = Float64(Float64(-0.037037037037037035 + Float64(a * Float64(a * a))) * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(Float64(a * a) * Float64(243.0 + Float64(a * -729.0)))))));
	elseif (rand <= 1.4e+154)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(1.0 / Float64(-0.3333333333333333 - a)) * Float64(Float64(0.012345679012345678 - Float64(Float64(a * a) * Float64(a * a))) * Float64(9.0 + Float64(a * Float64(a * Float64(-81.0 + Float64(Float64(a * a) * 729.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.85e+126)
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))));
	elseif (rand <= 1.4e+154)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (1.0 / (-0.3333333333333333 - a)) * ((0.012345679012345678 - ((a * a) * (a * a))) * (9.0 + (a * (a * (-81.0 + ((a * a) * 729.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.85e+126], N[(N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(9.0 + N[(a * N[(-27.0 + N[(N[(a * a), $MachinePrecision] * N[(243.0 + N[(a * -729.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.4e+154], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(1.0 / N[(-0.3333333333333333 - a), $MachinePrecision]), $MachinePrecision] * N[(N[(0.012345679012345678 - N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(9.0 + N[(a * N[(a * N[(-81.0 + N[(N[(a * a), $MachinePrecision] * 729.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.8499999999999999e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 0.2%

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

   9. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + a \cdot \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) - 27\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(a \cdot \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) - 27\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2} \cdot \left(243 + -729 \cdot a\right) - 27\right)}\right)\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right)}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) + -27\right)\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + \color{blue}{{a}^{2} \cdot \left(243 + -729 \cdot a\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \color{blue}{\left({a}^{2} \cdot \left(243 + -729 \cdot a\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(243 + -729 \cdot a\right)}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{243} + -729 \cdot a\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{243} + -729 \cdot a\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(243, \color{blue}{\left(-729 \cdot a\right)}\right)\right)\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(243, \left(a \cdot \color{blue}{-729}\right)\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 49.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(243, \mathsf{*.f64}\left(a, \color{blue}{-729}\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 49.7%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot \left(-27 + \left(a \cdot a\right) \cdot \left(243 + a \cdot -729\right)\right)\right)} \]

   if -1.8499999999999999e126 < rand < 1.4e154

   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 84.2%

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

   7. Simplified 84.2%

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

   if 1.4e154 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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 44.5%

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

   9. Applied egg-rr 44.5%

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

       1. flip-- N/A

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

       2. div-inv N/A

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

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

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

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

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{\frac{1}{9} + a \cdot a}\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \left(\frac{1}{9} + a \cdot a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       11. *-lowering-*.f64 19.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   11. Applied egg-rr 19.8%

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

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + {a}^{2} \cdot \left(729 \cdot {a}^{2} - 81\right)\right)}\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left({a}^{2} \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left(\left(a \cdot a\right) \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left(a \cdot \left(a \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(a \cdot \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(729 \cdot {a}^{2} - 81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(729 \cdot {a}^{2} + \left(\mathsf{neg}\left(81\right)\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(729 \cdot {a}^{2} + -81\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(-81 + 729 \cdot {a}^{2}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \left(729 \cdot {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \left({a}^{2} \cdot 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\left({a}^{2}\right), 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\left(a \cdot a\right), 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

       13. *-lowering-*.f64 62.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{81}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-81, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), 729\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right)\right) \]

   14. Simplified 62.8%

       \[\leadsto \left(\left(0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot \left(a \cdot \left(-81 + \left(a \cdot a\right) \cdot 729\right)\right)\right)}\right) \cdot \frac{1}{-0.3333333333333333 - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.85 \cdot 10^{+126}:\\ \;\;\;\;\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot \left(-27 + \left(a \cdot a\right) \cdot \left(243 + a \cdot -729\right)\right)\right)\\ \mathbf{elif}\;rand \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 - a} \cdot \left(\left(0.012345679012345678 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot \left(a \cdot \left(-81 + \left(a \cdot a\right) \cdot 729\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.037037037037037035 + a \cdot \left(a \cdot a\right)\\ \mathbf{if}\;rand \leq -1.85 \cdot 10^{+126}:\\ \;\;\;\;t\_0 \cdot \left(9 + a \cdot \left(-27 + \left(a \cdot a\right) \cdot \left(243 + a \cdot -729\right)\right)\right)\\ \mathbf{elif}\;rand \leq 2.65 \cdot 10^{+154}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(9 + a \cdot \left(-27 + a \cdot \left(a \cdot 243\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (+ -0.037037037037037035 (* a (* a a)))))
   (if (<= rand -1.85e+126)
     (* t_0 (+ 9.0 (* a (+ -27.0 (* (* a a) (+ 243.0 (* a -729.0)))))))
     (if (<= rand 2.65e+154)
       (+ -0.3333333333333333 a)
       (* t_0 (+ 9.0 (* a (+ -27.0 (* a (* a 243.0))))))))))

C

double code(double a, double rand) {
	double t_0 = -0.037037037037037035 + (a * (a * a));
	double tmp;
	if (rand <= -1.85e+126) {
		tmp = t_0 * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))));
	} else if (rand <= 2.65e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	}
	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 = (-0.037037037037037035d0) + (a * (a * a))
    if (rand <= (-1.85d+126)) then
        tmp = t_0 * (9.0d0 + (a * ((-27.0d0) + ((a * a) * (243.0d0 + (a * (-729.0d0)))))))
    else if (rand <= 2.65d+154) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0 * (9.0d0 + (a * ((-27.0d0) + (a * (a * 243.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = -0.037037037037037035 + (a * (a * a));
	double tmp;
	if (rand <= -1.85e+126) {
		tmp = t_0 * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))));
	} else if (rand <= 2.65e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = -0.037037037037037035 + (a * (a * a))
	tmp = 0
	if rand <= -1.85e+126:
		tmp = t_0 * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))))
	elif rand <= 2.65e+154:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(-0.037037037037037035 + Float64(a * Float64(a * a)))
	tmp = 0.0
	if (rand <= -1.85e+126)
		tmp = Float64(t_0 * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(Float64(a * a) * Float64(243.0 + Float64(a * -729.0)))))));
	elseif (rand <= 2.65e+154)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(t_0 * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * Float64(a * 243.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = -0.037037037037037035 + (a * (a * a));
	tmp = 0.0;
	if (rand <= -1.85e+126)
		tmp = t_0 * (9.0 + (a * (-27.0 + ((a * a) * (243.0 + (a * -729.0))))));
	elseif (rand <= 2.65e+154)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.85e+126], N[(t$95$0 * N[(9.0 + N[(a * N[(-27.0 + N[(N[(a * a), $MachinePrecision] * N[(243.0 + N[(a * -729.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 2.65e+154], N[(-0.3333333333333333 + a), $MachinePrecision], N[(t$95$0 * N[(9.0 + N[(a * N[(-27.0 + N[(a * N[(a * 243.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.8499999999999999e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 0.2%

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

   9. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + a \cdot \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) - 27\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(a \cdot \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) - 27\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2} \cdot \left(243 + -729 \cdot a\right) - 27\right)}\right)\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(27\right)\right)}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \left(243 + -729 \cdot a\right) + -27\right)\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + \color{blue}{{a}^{2} \cdot \left(243 + -729 \cdot a\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \color{blue}{\left({a}^{2} \cdot \left(243 + -729 \cdot a\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(243 + -729 \cdot a\right)}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{243} + -729 \cdot a\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{243} + -729 \cdot a\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(243, \color{blue}{\left(-729 \cdot a\right)}\right)\right)\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(243, \left(a \cdot \color{blue}{-729}\right)\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 49.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(243, \mathsf{*.f64}\left(a, \color{blue}{-729}\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 49.7%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot \left(-27 + \left(a \cdot a\right) \cdot \left(243 + a \cdot -729\right)\right)\right)} \]

   if -1.8499999999999999e126 < rand < 2.65000000000000012e154

   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 84.2%

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

   7. Simplified 84.2%

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

   if 2.65000000000000012e154 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 11.9%

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

   9. Applied egg-rr 11.9%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + a \cdot \left(243 \cdot {a}^{2} - 27\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(a \cdot \left(243 \cdot {a}^{2} - 27\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left(243 \cdot {a}^{2} - 27\right)}\right)\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(243 \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(27\right)\right)}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(243 \cdot {a}^{2} + -27\right)\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + \color{blue}{243 \cdot {a}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \color{blue}{\left(243 \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left({a}^{2} \cdot \color{blue}{243}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(\left(a \cdot a\right) \cdot 243\right)\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(a \cdot \color{blue}{\left(a \cdot 243\right)}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot 243\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 62.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{243}\right)\right)\right)\right)\right)\right) \]

   12. Simplified 62.8%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot \left(-27 + a \cdot \left(a \cdot 243\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.4 \cdot 10^{+126}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3\right)\\ \mathbf{elif}\;rand \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot \left(-27 + a \cdot \left(a \cdot 243\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.4e+126)
   (*
    (- 0.1111111111111111 (* a a))
    (+ (* a (+ 9.0 (* a (+ -27.0 (* a 81.0))))) -3.0))
   (if (<= rand 1.1e+154)
     (+ -0.3333333333333333 a)
     (*
      (+ -0.037037037037037035 (* a (* a a)))
      (+ 9.0 (* a (+ -27.0 (* a (* a 243.0)))))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.4e+126) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	} else if (rand <= 1.1e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.4d+126)) then
        tmp = (0.1111111111111111d0 - (a * a)) * ((a * (9.0d0 + (a * ((-27.0d0) + (a * 81.0d0))))) + (-3.0d0))
    else if (rand <= 1.1d+154) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = ((-0.037037037037037035d0) + (a * (a * a))) * (9.0d0 + (a * ((-27.0d0) + (a * (a * 243.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.4e+126) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	} else if (rand <= 1.1e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.4e+126:
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0)
	elif rand <= 1.1e+154:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + (a * (a * 243.0)))))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.4e+126)
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(Float64(a * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * 81.0))))) + -3.0));
	elseif (rand <= 1.1e+154)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(-0.037037037037037035 + Float64(a * Float64(a * a))) * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * Float64(a * 243.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.4e+126)
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	elseif (rand <= 1.1e+154)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (-0.037037037037037035 + (a * (a * a))) * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.4e+126], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(9.0 + N[(a * N[(-27.0 + N[(a * 81.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.1e+154], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(9.0 + N[(a * N[(-27.0 + N[(a * N[(a * 243.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.40000000000000005e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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 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. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot \left(81 \cdot a - 27\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a - 27\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + \left(\mathsf{neg}\left(27\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + -27\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + 81 \cdot a\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(81 \cdot a\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(a \cdot 81\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, 81\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       12. metadata-eval 49.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, 81\right)\right)\right)\right)\right), -3\right)\right) \]

   12. Simplified 49.6%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3\right)} \]

   if -1.40000000000000005e126 < rand < 1.1000000000000001e154

   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 84.2%

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

   7. Simplified 84.2%

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

   if 1.1000000000000001e154 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 11.9%

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

   9. Applied egg-rr 11.9%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + a \cdot \left(243 \cdot {a}^{2} - 27\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(a \cdot \left(243 \cdot {a}^{2} - 27\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left(243 \cdot {a}^{2} - 27\right)}\right)\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(243 \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(27\right)\right)}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(243 \cdot {a}^{2} + -27\right)\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + \color{blue}{243 \cdot {a}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \color{blue}{\left(243 \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left({a}^{2} \cdot \color{blue}{243}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(\left(a \cdot a\right) \cdot 243\right)\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(a \cdot \color{blue}{\left(a \cdot 243\right)}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot 243\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 62.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{243}\right)\right)\right)\right)\right)\right) \]

   12. Simplified 62.8%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot \left(-27 + a \cdot \left(a \cdot 243\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 74.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.037037037037037035 + a \cdot \left(a \cdot a\right)\\ \mathbf{if}\;rand \leq -2.05 \cdot 10^{+126}:\\ \;\;\;\;t\_0 \cdot \left(9 + a \cdot -27\right)\\ \mathbf{elif}\;rand \leq 3.2 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(9 + a \cdot \left(-27 + a \cdot \left(a \cdot 243\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (+ -0.037037037037037035 (* a (* a a)))))
   (if (<= rand -2.05e+126)
     (* t_0 (+ 9.0 (* a -27.0)))
     (if (<= rand 3.2e+153)
       (+ -0.3333333333333333 a)
       (* t_0 (+ 9.0 (* a (+ -27.0 (* a (* a 243.0))))))))))

C

double code(double a, double rand) {
	double t_0 = -0.037037037037037035 + (a * (a * a));
	double tmp;
	if (rand <= -2.05e+126) {
		tmp = t_0 * (9.0 + (a * -27.0));
	} else if (rand <= 3.2e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	}
	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 = (-0.037037037037037035d0) + (a * (a * a))
    if (rand <= (-2.05d+126)) then
        tmp = t_0 * (9.0d0 + (a * (-27.0d0)))
    else if (rand <= 3.2d+153) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0 * (9.0d0 + (a * ((-27.0d0) + (a * (a * 243.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = -0.037037037037037035 + (a * (a * a));
	double tmp;
	if (rand <= -2.05e+126) {
		tmp = t_0 * (9.0 + (a * -27.0));
	} else if (rand <= 3.2e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = -0.037037037037037035 + (a * (a * a))
	tmp = 0
	if rand <= -2.05e+126:
		tmp = t_0 * (9.0 + (a * -27.0))
	elif rand <= 3.2e+153:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(-0.037037037037037035 + Float64(a * Float64(a * a)))
	tmp = 0.0
	if (rand <= -2.05e+126)
		tmp = Float64(t_0 * Float64(9.0 + Float64(a * -27.0)));
	elseif (rand <= 3.2e+153)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(t_0 * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * Float64(a * 243.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = -0.037037037037037035 + (a * (a * a));
	tmp = 0.0;
	if (rand <= -2.05e+126)
		tmp = t_0 * (9.0 + (a * -27.0));
	elseif (rand <= 3.2e+153)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0 * (9.0 + (a * (-27.0 + (a * (a * 243.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -2.05e+126], N[(t$95$0 * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 3.2e+153], N[(-0.3333333333333333 + a), $MachinePrecision], N[(t$95$0 * N[(9.0 + N[(a * N[(-27.0 + N[(a * N[(a * 243.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.05e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 0.2%

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

   9. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + -27 \cdot a\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(-27 \cdot a\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left(a \cdot \color{blue}{-27}\right)\right)\right) \]

       3. *-lowering-*.f64 46.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{-27}\right)\right)\right) \]

   12. Simplified 46.6%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot -27\right)} \]

   if -2.05e126 < rand < 3.2000000000000001e153

   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 84.2%

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

   7. Simplified 84.2%

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

   if 3.2000000000000001e153 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 11.9%

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

   9. Applied egg-rr 11.9%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + a \cdot \left(243 \cdot {a}^{2} - 27\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(a \cdot \left(243 \cdot {a}^{2} - 27\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left(243 \cdot {a}^{2} - 27\right)}\right)\right)\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(243 \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(27\right)\right)}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(243 \cdot {a}^{2} + -27\right)\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + \color{blue}{243 \cdot {a}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \color{blue}{\left(243 \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left({a}^{2} \cdot \color{blue}{243}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(\left(a \cdot a\right) \cdot 243\right)\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(a \cdot \color{blue}{\left(a \cdot 243\right)}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot 243\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 62.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{243}\right)\right)\right)\right)\right)\right) \]

   12. Simplified 62.8%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot \left(-27 + a \cdot \left(a \cdot 243\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 74.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 9 + a \cdot -27\\ \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot t\_0\\ \mathbf{elif}\;rand \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (+ 9.0 (* a -27.0))))
   (if (<= rand -2.25e+126)
     (* (+ -0.037037037037037035 (* a (* a a))) t_0)
     (if (<= rand 2.1e+153)
       (+ -0.3333333333333333 a)
       (* (- 0.1111111111111111 (* a a)) (+ -3.0 (* a t_0)))))))

C

double code(double a, double rand) {
	double t_0 = 9.0 + (a * -27.0);
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = (-0.037037037037037035 + (a * (a * a))) * t_0;
	} else if (rand <= 2.1e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 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 = 9.0d0 + (a * (-27.0d0))
    if (rand <= (-2.25d+126)) then
        tmp = ((-0.037037037037037035d0) + (a * (a * a))) * t_0
    else if (rand <= 2.1d+153) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (0.1111111111111111d0 - (a * a)) * ((-3.0d0) + (a * t_0))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 9.0 + (a * -27.0);
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = (-0.037037037037037035 + (a * (a * a))) * t_0;
	} else if (rand <= 2.1e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * t_0));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 9.0 + (a * -27.0)
	tmp = 0
	if rand <= -2.25e+126:
		tmp = (-0.037037037037037035 + (a * (a * a))) * t_0
	elif rand <= 2.1e+153:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * t_0))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(9.0 + Float64(a * -27.0))
	tmp = 0.0
	if (rand <= -2.25e+126)
		tmp = Float64(Float64(-0.037037037037037035 + Float64(a * Float64(a * a))) * t_0);
	elseif (rand <= 2.1e+153)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(-3.0 + Float64(a * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 9.0 + (a * -27.0);
	tmp = 0.0;
	if (rand <= -2.25e+126)
		tmp = (-0.037037037037037035 + (a * (a * a))) * t_0;
	elseif (rand <= 2.1e+153)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -2.25e+126], N[(N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[rand, 2.1e+153], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-3.0 + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.24999999999999987e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 0.2%

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

   9. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + -27 \cdot a\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(-27 \cdot a\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left(a \cdot \color{blue}{-27}\right)\right)\right) \]

       3. *-lowering-*.f64 46.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{-27}\right)\right)\right) \]

   12. Simplified 46.6%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot -27\right)} \]

   if -2.24999999999999987e126 < rand < 2.10000000000000017e153

   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 84.2%

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

   7. Simplified 84.2%

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

   if 2.10000000000000017e153 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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 44.5%

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

   9. Applied egg-rr 44.5%

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

   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. +-lowering-+.f64 N/A

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

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

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

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

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

       5. *-commutative N/A

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

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

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

       7. metadata-eval 62.7%

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

   12. Simplified 62.7%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot -27\right)\\ \mathbf{elif}\;rand \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.037037037037037035 + a \cdot \left(a \cdot a\right)\\ \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;t\_0 \cdot \left(9 + a \cdot -27\right)\\ \mathbf{elif}\;rand \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;9 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (+ -0.037037037037037035 (* a (* a a)))))
   (if (<= rand -2.25e+126)
     (* t_0 (+ 9.0 (* a -27.0)))
     (if (<= rand 1.7e+153) (+ -0.3333333333333333 a) (* 9.0 t_0)))))

C

double code(double a, double rand) {
	double t_0 = -0.037037037037037035 + (a * (a * a));
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = t_0 * (9.0 + (a * -27.0));
	} else if (rand <= 1.7e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 9.0 * 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 = (-0.037037037037037035d0) + (a * (a * a))
    if (rand <= (-2.25d+126)) then
        tmp = t_0 * (9.0d0 + (a * (-27.0d0)))
    else if (rand <= 1.7d+153) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = 9.0d0 * t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = -0.037037037037037035 + (a * (a * a));
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = t_0 * (9.0 + (a * -27.0));
	} else if (rand <= 1.7e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 9.0 * t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = -0.037037037037037035 + (a * (a * a))
	tmp = 0
	if rand <= -2.25e+126:
		tmp = t_0 * (9.0 + (a * -27.0))
	elif rand <= 1.7e+153:
		tmp = -0.3333333333333333 + a
	else:
		tmp = 9.0 * t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(-0.037037037037037035 + Float64(a * Float64(a * a)))
	tmp = 0.0
	if (rand <= -2.25e+126)
		tmp = Float64(t_0 * Float64(9.0 + Float64(a * -27.0)));
	elseif (rand <= 1.7e+153)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(9.0 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = -0.037037037037037035 + (a * (a * a));
	tmp = 0.0;
	if (rand <= -2.25e+126)
		tmp = t_0 * (9.0 + (a * -27.0));
	elseif (rand <= 1.7e+153)
		tmp = -0.3333333333333333 + a;
	else
		tmp = 9.0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -2.25e+126], N[(t$95$0 * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.7e+153], N[(-0.3333333333333333 + a), $MachinePrecision], N[(9.0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.24999999999999987e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 0.2%

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

   9. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{\left(9 + -27 \cdot a\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \color{blue}{\left(-27 \cdot a\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \left(a \cdot \color{blue}{-27}\right)\right)\right) \]

       3. *-lowering-*.f64 46.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{-27}\right)\right)\right) \]

   12. Simplified 46.6%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(9 + a \cdot -27\right)} \]

   if -2.24999999999999987e126 < rand < 1.6999999999999999e153

   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 84.2%

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

   7. Simplified 84.2%

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

   if 1.6999999999999999e153 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 11.9%

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

   9. Applied egg-rr 11.9%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{9}\right) \]
   11. Step-by-step derivation

   12. Simplified 55.4%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{9} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \left(9 + a \cdot -27\right)\\ \mathbf{elif}\;rand \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot 9\right)\\ \mathbf{elif}\;rand \leq 2.5 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -2.25e+126)
   (* (- 0.1111111111111111 (* a a)) (+ -3.0 (* a 9.0)))
   (if (<= rand 2.5e+153)
     (+ -0.3333333333333333 a)
     (* 9.0 (+ -0.037037037037037035 (* a (* a a)))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0));
	} else if (rand <= 2.5e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 9.0 * (-0.037037037037037035 + (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 <= (-2.25d+126)) then
        tmp = (0.1111111111111111d0 - (a * a)) * ((-3.0d0) + (a * 9.0d0))
    else if (rand <= 2.5d+153) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = 9.0d0 * ((-0.037037037037037035d0) + (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -2.25e+126) {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0));
	} else if (rand <= 2.5e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 9.0 * (-0.037037037037037035 + (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -2.25e+126:
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0))
	elif rand <= 2.5e+153:
		tmp = -0.3333333333333333 + a
	else:
		tmp = 9.0 * (-0.037037037037037035 + (a * (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -2.25e+126)
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(-3.0 + Float64(a * 9.0)));
	elseif (rand <= 2.5e+153)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(9.0 * Float64(-0.037037037037037035 + Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -2.25e+126)
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0));
	elseif (rand <= 2.5e+153)
		tmp = -0.3333333333333333 + a;
	else
		tmp = 9.0 * (-0.037037037037037035 + (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -2.25e+126], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-3.0 + N[(a * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 2.5e+153], N[(-0.3333333333333333 + a), $MachinePrecision], N[(9.0 * N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.24999999999999987e126

   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 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 0.3%

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

   7. Simplified 0.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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 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. +-lowering-+.f64 N/A

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

       3. *-commutative N/A

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

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

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

       5. metadata-eval 40.6%

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

   12. Simplified 40.6%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot 9 + -3\right)} \]

   if -2.24999999999999987e126 < rand < 2.50000000000000009e153

   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 84.2%

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

   7. Simplified 84.2%

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

   if 2.50000000000000009e153 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 11.9%

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

   9. Applied egg-rr 11.9%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{9}\right) \]
   11. Step-by-step derivation

   12. Simplified 55.4%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{9} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.25 \cdot 10^{+126}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot 9\right)\\ \mathbf{elif}\;rand \leq 2.5 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 1.85 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 1.85e+153)
   (+ -0.3333333333333333 a)
   (* 9.0 (+ -0.037037037037037035 (* a (* a a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 1.85e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 9.0 * (-0.037037037037037035 + (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 <= 1.85d+153) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = 9.0d0 * ((-0.037037037037037035d0) + (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 1.85e+153) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 9.0 * (-0.037037037037037035 + (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 1.85e+153:
		tmp = -0.3333333333333333 + a
	else:
		tmp = 9.0 * (-0.037037037037037035 + (a * (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 1.85e+153)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(9.0 * Float64(-0.037037037037037035 + Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 1.85e+153)
		tmp = -0.3333333333333333 + a;
	else
		tmp = 9.0 * (-0.037037037037037035 + (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 1.85e+153], N[(-0.3333333333333333 + a), $MachinePrecision], N[(9.0 * N[(-0.037037037037037035 + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 1.8500000000000001e153

   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 72.2%

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

   7. Simplified 72.2%

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

   if 1.8500000000000001e153 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. div-inv N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. cube-mult N/A

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

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

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

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

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

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

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

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

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

      13. metadata-eval N/A

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

      14. distribute-rgt-out-- N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 11.9%

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

   9. Applied egg-rr 11.9%

      \[\leadsto \color{blue}{\left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \frac{1}{0.1111111111111111 + a \cdot \left(a + 0.3333333333333333\right)}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{27}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right), \color{blue}{9}\right) \]
   11. Step-by-step derivation

   12. Simplified 55.4%

       \[\leadsto \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 1.85 \cdot 10^{+153}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(-0.037037037037037035 + a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 2.9 \cdot 10^{+148}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 2.9e+148)
   (+ -0.3333333333333333 a)
   (* (- 0.1111111111111111 (* a a)) -3.0)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 2.9e+148) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 2.9d+148) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (0.1111111111111111d0 - (a * a)) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 2.9e+148) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 2.9e+148:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (0.1111111111111111 - (a * a)) * -3.0
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 2.9e+148)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 2.9e+148)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 2.9e+148], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 2.9e148

   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 72.5%

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

   7. Simplified 72.5%

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

   if 2.9e148 < rand

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 5.9%

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

   7. Simplified 5.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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 43.5%

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

   9. Applied egg-rr 43.5%

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

   10. Taylor expanded in a around 0

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

   12. Simplified 44.8%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{-3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.1% accurate, 39.7× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 + a \end{array} \]

FPCore

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

C

double code(double a, double rand) {
	return -0.3333333333333333 + a;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (-0.3333333333333333d0) + a
end function

Java

public static double code(double a, double rand) {
	return -0.3333333333333333 + a;
}

Python

def code(a, rand):
	return -0.3333333333333333 + a

Julia

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

MATLAB

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

Wolfram

code[a_, rand_] := N[(-0.3333333333333333 + a), $MachinePrecision]

Derivation

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 62.1%

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

7. Simplified 62.1%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
8. Add Preprocessing

Alternative 21: 61.1% 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 99.9%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
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 62.1%

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

7. Simplified 62.1%

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

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 22: 1.6% 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 99.9%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
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 62.1%

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

7. Simplified 62.1%

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

8. Taylor expanded in a around 0

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

10. Simplified 1.4%

    \[\leadsto \color{blue}{-0.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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))))