Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 12.5s

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

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

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(Float64(sqrt(Float64(-0.3333333333333333 + a)) * 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[(N[Sqrt[N[(-0.3333333333333333 + a), $MachinePrecision]], $MachinePrecision] * rand), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

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

   \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]
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 99.8%

   \[\leadsto \color{blue}{\sqrt{-0.3333333333333333 + a} \cdot \left(0.3333333333333333 \cdot rand\right) + \left(-0.3333333333333333 + a\right)} \]
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 99.8%

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

9. Applied egg-rr 99.8%

   \[\leadsto \sqrt{-0.3333333333333333 + a} \cdot \color{blue}{\frac{rand}{3}} + \left(-0.3333333333333333 + a\right) \]
10. Step-by-step derivation

    1. associate-*r/ N/A

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

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

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

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

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

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

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

    5. +-lowering-+.f64 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-0.3333333333333333 + a} \cdot \frac{rand}{3}\\ \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ -0.3333333333333333 a)) (/ rand 3.0))))
   (if (<= rand -1.4e+42)
     t_0
     (if (<= rand 8.5e+56) (+ -0.3333333333333333 a) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((-0.3333333333333333 + a)) * (rand / 3.0);
	double tmp;
	if (rand <= -1.4e+42) {
		tmp = t_0;
	} else if (rand <= 8.5e+56) {
		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 / 3.0d0)
    if (rand <= (-1.4d+42)) then
        tmp = t_0
    else if (rand <= 8.5d+56) 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 / 3.0);
	double tmp;
	if (rand <= -1.4e+42) {
		tmp = t_0;
	} else if (rand <= 8.5e+56) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((-0.3333333333333333 + a)) * (rand / 3.0)
	tmp = 0
	if rand <= -1.4e+42:
		tmp = t_0
	elif rand <= 8.5e+56:
		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 / 3.0))
	tmp = 0.0
	if (rand <= -1.4e+42)
		tmp = t_0;
	elseif (rand <= 8.5e+56)
		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 / 3.0);
	tmp = 0.0;
	if (rand <= -1.4e+42)
		tmp = t_0;
	elseif (rand <= 8.5e+56)
		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 / 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.4e+42], t$95$0, If[LessEqual[rand, 8.5e+56], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.4e42 or 8.4999999999999998e56 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.5%

      \[\leadsto \left(a + -0.3333333333333333\right) \cdot \left(1 + \color{blue}{\frac{{\left(a + -0.3333333333333333\right)}^{-0.5}}{3} \cdot rand}\right) \]

   7. Taylor expanded in rand around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 91.1%

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

   9. Simplified 91.1%

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. div-inv N/A

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

       4. /-lowering-/.f64 91.2%

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

   11. Applied egg-rr 91.2%

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

   if -1.4e42 < rand < 8.4999999999999998e56

   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 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{-0.3333333333333333 + a} \cdot \frac{rand}{3}\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.3333333333333333 + a} \cdot \frac{rand}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-0.3333333333333333 + a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ -0.3333333333333333 a)) (* rand 0.3333333333333333))))
   (if (<= rand -1.4e+42)
     t_0
     (if (<= rand 8.5e+56) (+ -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 <= -1.4e+42) {
		tmp = t_0;
	} else if (rand <= 8.5e+56) {
		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 <= (-1.4d+42)) then
        tmp = t_0
    else if (rand <= 8.5d+56) 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 <= -1.4e+42) {
		tmp = t_0;
	} else if (rand <= 8.5e+56) {
		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 <= -1.4e+42:
		tmp = t_0
	elif rand <= 8.5e+56:
		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 <= -1.4e+42)
		tmp = t_0;
	elseif (rand <= 8.5e+56)
		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 <= -1.4e+42)
		tmp = t_0;
	elseif (rand <= 8.5e+56)
		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, -1.4e+42], t$95$0, If[LessEqual[rand, 8.5e+56], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.4e42 or 8.4999999999999998e56 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 91.1%

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

   7. Simplified 91.1%

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

   if -1.4e42 < rand < 8.4999999999999998e56

   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 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{-0.3333333333333333 + a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.3333333333333333 + a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.4e+42)
   (* rand (* 0.3333333333333333 (sqrt a)))
   (if (<= rand 8.5e+56)
     (+ -0.3333333333333333 a)
     (* (* rand 0.3333333333333333) (sqrt a)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.4e+42) {
		tmp = rand * (0.3333333333333333 * sqrt(a));
	} else if (rand <= 8.5e+56) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (rand * 0.3333333333333333) * 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 <= (-1.4d+42)) then
        tmp = rand * (0.3333333333333333d0 * sqrt(a))
    else if (rand <= 8.5d+56) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (rand * 0.3333333333333333d0) * sqrt(a)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.4e+42) {
		tmp = rand * (0.3333333333333333 * Math.sqrt(a));
	} else if (rand <= 8.5e+56) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (rand * 0.3333333333333333) * Math.sqrt(a);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.4e+42:
		tmp = rand * (0.3333333333333333 * math.sqrt(a))
	elif rand <= 8.5e+56:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (rand * 0.3333333333333333) * math.sqrt(a)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.4e+42)
		tmp = Float64(rand * Float64(0.3333333333333333 * sqrt(a)));
	elseif (rand <= 8.5e+56)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(rand * 0.3333333333333333) * sqrt(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.4e+42)
		tmp = rand * (0.3333333333333333 * sqrt(a));
	elseif (rand <= 8.5e+56)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.4e+42], N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 8.5e+56], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.4e42

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.6%

      \[\leadsto \left(a + -0.3333333333333333\right) \cdot \left(1 + \color{blue}{\frac{{\left(a + -0.3333333333333333\right)}^{-0.5}}{3} \cdot rand}\right) \]

   7. Taylor expanded in rand around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 94.0%

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

   9. Simplified 94.0%

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

   10. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 92.5%

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

   12. Simplified 92.5%

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

   if -1.4e42 < rand < 8.4999999999999998e56

   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 98.1%

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

   7. Simplified 98.1%

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

   if 8.4999999999999998e56 < rand

   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. Step-by-step derivation

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.5%

      \[\leadsto \left(a + -0.3333333333333333\right) \cdot \left(1 + \color{blue}{\frac{{\left(a + -0.3333333333333333\right)}^{-0.5}}{3} \cdot rand}\right) \]

   7. Taylor expanded in rand around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 88.1%

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

   9. Simplified 88.1%

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

   10. Taylor expanded in a around inf

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

       1. sqrt-lowering-sqrt.f64 86.2%

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

   12. Simplified 86.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.4e+42)
   (* rand (* 0.3333333333333333 (sqrt a)))
   (if (<= rand 8.5e+56)
     (+ -0.3333333333333333 a)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.4e+42) {
		tmp = rand * (0.3333333333333333 * sqrt(a));
	} else if (rand <= 8.5e+56) {
		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 <= (-1.4d+42)) then
        tmp = rand * (0.3333333333333333d0 * sqrt(a))
    else if (rand <= 8.5d+56) 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 <= -1.4e+42) {
		tmp = rand * (0.3333333333333333 * Math.sqrt(a));
	} else if (rand <= 8.5e+56) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.4e+42:
		tmp = rand * (0.3333333333333333 * math.sqrt(a))
	elif rand <= 8.5e+56:
		tmp = -0.3333333333333333 + a
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.4e+42)
		tmp = Float64(rand * Float64(0.3333333333333333 * sqrt(a)));
	elseif (rand <= 8.5e+56)
		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 <= -1.4e+42)
		tmp = rand * (0.3333333333333333 * sqrt(a));
	elseif (rand <= 8.5e+56)
		tmp = -0.3333333333333333 + a;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.4e+42], N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 8.5e+56], 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 < -1.4e42

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.6%

      \[\leadsto \left(a + -0.3333333333333333\right) \cdot \left(1 + \color{blue}{\frac{{\left(a + -0.3333333333333333\right)}^{-0.5}}{3} \cdot rand}\right) \]

   7. Taylor expanded in rand around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 94.0%

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

   9. Simplified 94.0%

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

   10. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 92.5%

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

   12. Simplified 92.5%

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

   if -1.4e42 < rand < 8.4999999999999998e56

   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 98.1%

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

   7. Simplified 98.1%

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

   if 8.4999999999999998e56 < rand

   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 a around inf

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

   7. Simplified 97.6%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 97.6%

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

   11. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)} \]
   12. 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 86.2%

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

   13. Simplified 86.2%

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

Alternative 6: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -1.4e+42)
     t_0
     (if (<= rand 8.5e+56) (+ -0.3333333333333333 a) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -1.4e+42) {
		tmp = t_0;
	} else if (rand <= 8.5e+56) {
		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 <= (-1.4d+42)) then
        tmp = t_0
    else if (rand <= 8.5d+56) 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 <= -1.4e+42) {
		tmp = t_0;
	} else if (rand <= 8.5e+56) {
		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 <= -1.4e+42:
		tmp = t_0
	elif rand <= 8.5e+56:
		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 <= -1.4e+42)
		tmp = t_0;
	elseif (rand <= 8.5e+56)
		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 <= -1.4e+42)
		tmp = t_0;
	elseif (rand <= 8.5e+56)
		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, -1.4e+42], t$95$0, If[LessEqual[rand, 8.5e+56], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.4e42 or 8.4999999999999998e56 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around inf

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

   7. Simplified 97.7%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 97.7%

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

   11. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)} \]
   12. 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 89.2%

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

   13. Simplified 89.2%

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

   if -1.4e42 < rand < 8.4999999999999998e56

   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 98.1%

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

   7. Simplified 98.1%

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

Alternative 7: 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 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
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 99.8%

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

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

   \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]
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 99.8%

   \[\leadsto \color{blue}{\sqrt{-0.3333333333333333 + a} \cdot \left(0.3333333333333333 \cdot rand\right) + \left(-0.3333333333333333 + a\right)} \]
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 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 8: 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 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
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 99.8%

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

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

   \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]
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 99.8%

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

8. Final simplification 99.8%

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

Alternative 9: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = ((-0.3333333333333333d0) + a) * (1.0d0 + ((rand / sqrt(a)) / 3.0d0))
end function

Java

public static double code(double a, double rand) {
	return (-0.3333333333333333 + a) * (1.0 + ((rand / Math.sqrt(a)) / 3.0));
}

Python

def code(a, rand):
	return (-0.3333333333333333 + a) * (1.0 + ((rand / math.sqrt(a)) / 3.0))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (-0.3333333333333333 + a) * (1.0 + ((rand / sqrt(a)) / 3.0));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

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 a around inf

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

7. Simplified 99.0%

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

   1. sqrt-prod N/A

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

   2. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{{a}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{9}}}\right)\right)\right) \]

   3. metadata-eval N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. pow1/2 N/A

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

   8. sqrt-lowering-sqrt.f64 99.0%

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

9. Applied egg-rr 99.0%

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

10. Final simplification 99.0%

    \[\leadsto \left(-0.3333333333333333 + a\right) \cdot \left(1 + \frac{\frac{rand}{\sqrt{a}}}{3}\right) \]
11. Add Preprocessing

Alternative 10: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

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 a around inf

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 11: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

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. Step-by-step derivation

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. distribute-lft1-in N/A

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

   13. associate-+r- N/A

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

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

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 12: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

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

   \[\leadsto \color{blue}{\left(a + \frac{1}{3} \cdot \left(rand \cdot \sqrt{a - \frac{1}{3}}\right)\right) - \frac{1}{3}} \]
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 99.8%

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

8. Taylor expanded in a around inf

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

10. Simplified 98.9%

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

11. Final simplification 98.9%

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

Alternative 13: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

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 a around inf

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

7. Simplified 99.0%

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

8. Taylor expanded in a around inf

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

10. Simplified 98.1%

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

Alternative 14: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

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 a around inf

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

7. Simplified 99.0%

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

8. Taylor expanded in a around inf

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

10. Simplified 98.1%

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

11. Taylor expanded in a around 0

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

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

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

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

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

    3. *-commutative N/A

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

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

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

    5. sqrt-lowering-sqrt.f64 98.1%

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

13. Simplified 98.1%

    \[\leadsto \color{blue}{a + 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)} \]
14. Add Preprocessing

Alternative 15: 73.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{0.1111111111111111 - a \cdot a}\\ \mathbf{if}\;rand \leq -8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3}{t\_0}\\ \mathbf{elif}\;rand \leq 1.85 \cdot 10^{+123}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + a \cdot \left(9 + a \cdot -27\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- 0.1111111111111111 (* a a)))))
   (if (<= rand -8.5e+76)
     (/ (+ (* a (+ 9.0 (* a (+ -27.0 (* a 81.0))))) -3.0) t_0)
     (if (<= rand 1.85e+123)
       (+ -0.3333333333333333 a)
       (/ (+ -3.0 (* a (+ 9.0 (* a -27.0)))) t_0)))))

C

double code(double a, double rand) {
	double t_0 = 1.0 / (0.1111111111111111 - (a * a));
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0) / t_0;
	} else if (rand <= 1.85e+123) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / 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 = 1.0d0 / (0.1111111111111111d0 - (a * a))
    if (rand <= (-8.5d+76)) then
        tmp = ((a * (9.0d0 + (a * ((-27.0d0) + (a * 81.0d0))))) + (-3.0d0)) / t_0
    else if (rand <= 1.85d+123) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 1.0 / (0.1111111111111111 - (a * a));
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0) / t_0;
	} else if (rand <= 1.85e+123) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 1.0 / (0.1111111111111111 - (a * a))
	tmp = 0
	if rand <= -8.5e+76:
		tmp = ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0) / t_0
	elif rand <= 1.85e+123:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(1.0 / Float64(0.1111111111111111 - Float64(a * a)))
	tmp = 0.0
	if (rand <= -8.5e+76)
		tmp = Float64(Float64(Float64(a * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * 81.0))))) + -3.0) / t_0);
	elseif (rand <= 1.85e+123)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 1.0 / (0.1111111111111111 - (a * a));
	tmp = 0.0;
	if (rand <= -8.5e+76)
		tmp = ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0) / t_0;
	elseif (rand <= 1.85e+123)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(1.0 / N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.5e+76], N[(N[(N[(a * N[(9.0 + N[(a * N[(-27.0 + N[(a * 81.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[rand, 1.85e+123], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -8.49999999999999992e76

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 0.6%

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

   7. Simplified 0.6%

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

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 0.5%

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

   9. Applied egg-rr 0.5%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 0.5%

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

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

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

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

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

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. metadata-eval 27.5%

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

   14. Simplified 27.5%

       \[\leadsto \frac{\color{blue}{a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]

   if -8.49999999999999992e76 < rand < 1.84999999999999998e123

   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 91.4%

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

   7. Simplified 91.4%

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

   if 1.84999999999999998e123 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 6.4%

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

   7. Simplified 6.4%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 29.7%

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

   9. Applied egg-rr 29.7%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 29.7%

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

   11. Applied egg-rr 29.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

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

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

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

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

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

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

       5. *-commutative N/A

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

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

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

       7. metadata-eval 33.6%

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

   14. Simplified 33.6%

       \[\leadsto \frac{\color{blue}{a \cdot \left(9 + a \cdot -27\right) + -3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3}{\frac{1}{0.1111111111111111 - a \cdot a}}\\ \mathbf{elif}\;rand \leq 1.85 \cdot 10^{+123}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + a \cdot \left(9 + a \cdot -27\right)}{\frac{1}{0.1111111111111111 - a \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{0.1111111111111111 - a \cdot a}\\ \mathbf{if}\;rand \leq -8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot 9 + -3}{t\_0}\\ \mathbf{elif}\;rand \leq 1.85 \cdot 10^{+123}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + a \cdot \left(9 + a \cdot -27\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- 0.1111111111111111 (* a a)))))
   (if (<= rand -8.5e+76)
     (/ (+ (* a 9.0) -3.0) t_0)
     (if (<= rand 1.85e+123)
       (+ -0.3333333333333333 a)
       (/ (+ -3.0 (* a (+ 9.0 (* a -27.0)))) t_0)))))

C

double code(double a, double rand) {
	double t_0 = 1.0 / (0.1111111111111111 - (a * a));
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * 9.0) + -3.0) / t_0;
	} else if (rand <= 1.85e+123) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / 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 = 1.0d0 / (0.1111111111111111d0 - (a * a))
    if (rand <= (-8.5d+76)) then
        tmp = ((a * 9.0d0) + (-3.0d0)) / t_0
    else if (rand <= 1.85d+123) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 1.0 / (0.1111111111111111 - (a * a));
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * 9.0) + -3.0) / t_0;
	} else if (rand <= 1.85e+123) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 1.0 / (0.1111111111111111 - (a * a))
	tmp = 0
	if rand <= -8.5e+76:
		tmp = ((a * 9.0) + -3.0) / t_0
	elif rand <= 1.85e+123:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(1.0 / Float64(0.1111111111111111 - Float64(a * a)))
	tmp = 0.0
	if (rand <= -8.5e+76)
		tmp = Float64(Float64(Float64(a * 9.0) + -3.0) / t_0);
	elseif (rand <= 1.85e+123)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 1.0 / (0.1111111111111111 - (a * a));
	tmp = 0.0;
	if (rand <= -8.5e+76)
		tmp = ((a * 9.0) + -3.0) / t_0;
	elseif (rand <= 1.85e+123)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (-3.0 + (a * (9.0 + (a * -27.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(1.0 / N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.5e+76], N[(N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[rand, 1.85e+123], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -8.49999999999999992e76

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 0.6%

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

   7. Simplified 0.6%

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

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 0.5%

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

   9. Applied egg-rr 0.5%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 0.5%

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

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

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

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

       3. *-commutative N/A

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

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

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

       5. metadata-eval 27.4%

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

   14. Simplified 27.4%

       \[\leadsto \frac{\color{blue}{a \cdot 9 + -3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]

   if -8.49999999999999992e76 < rand < 1.84999999999999998e123

   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 91.4%

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

   7. Simplified 91.4%

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

   if 1.84999999999999998e123 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 6.4%

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

   7. Simplified 6.4%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 29.7%

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

   9. Applied egg-rr 29.7%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 29.7%

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

   11. Applied egg-rr 29.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

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

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

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

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

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

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

       5. *-commutative N/A

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

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

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

       7. metadata-eval 33.6%

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

   14. Simplified 33.6%

       \[\leadsto \frac{\color{blue}{a \cdot \left(9 + a \cdot -27\right) + -3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot 9 + -3}{\frac{1}{0.1111111111111111 - a \cdot a}}\\ \mathbf{elif}\;rand \leq 1.85 \cdot 10^{+123}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + a \cdot \left(9 + a \cdot -27\right)}{\frac{1}{0.1111111111111111 - a \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot 9 + -3}{\frac{1}{0.1111111111111111 - a \cdot a}}\\ \mathbf{elif}\;rand \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0013717421124828531 - \left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{-0.00411522633744856}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -8.5e+76)
   (/ (+ (* a 9.0) -3.0) (/ 1.0 (- 0.1111111111111111 (* a a))))
   (if (<= rand 2.3e+154)
     (+ -0.3333333333333333 a)
     (/
      (- 0.0013717421124828531 (* (* a a) (* a (* a (* a a)))))
      -0.00411522633744856))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * 9.0) + -3.0) / (1.0 / (0.1111111111111111 - (a * a)));
	} else if (rand <= 2.3e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.0013717421124828531 - ((a * a) * (a * (a * (a * a))))) / -0.00411522633744856;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-8.5d+76)) then
        tmp = ((a * 9.0d0) + (-3.0d0)) / (1.0d0 / (0.1111111111111111d0 - (a * a)))
    else if (rand <= 2.3d+154) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (0.0013717421124828531d0 - ((a * a) * (a * (a * (a * a))))) / (-0.00411522633744856d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * 9.0) + -3.0) / (1.0 / (0.1111111111111111 - (a * a)));
	} else if (rand <= 2.3e+154) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.0013717421124828531 - ((a * a) * (a * (a * (a * a))))) / -0.00411522633744856;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -8.5e+76:
		tmp = ((a * 9.0) + -3.0) / (1.0 / (0.1111111111111111 - (a * a)))
	elif rand <= 2.3e+154:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (0.0013717421124828531 - ((a * a) * (a * (a * (a * a))))) / -0.00411522633744856
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -8.5e+76)
		tmp = Float64(Float64(Float64(a * 9.0) + -3.0) / Float64(1.0 / Float64(0.1111111111111111 - Float64(a * a))));
	elseif (rand <= 2.3e+154)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(0.0013717421124828531 - Float64(Float64(a * a) * Float64(a * Float64(a * Float64(a * a))))) / -0.00411522633744856);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -8.5e+76)
		tmp = ((a * 9.0) + -3.0) / (1.0 / (0.1111111111111111 - (a * a)));
	elseif (rand <= 2.3e+154)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (0.0013717421124828531 - ((a * a) * (a * (a * (a * a))))) / -0.00411522633744856;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -8.5e+76], N[(N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision] / N[(1.0 / N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 2.3e+154], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(0.0013717421124828531 - N[(N[(a * a), $MachinePrecision] * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -0.00411522633744856), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -8.49999999999999992e76

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 0.6%

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

   7. Simplified 0.6%

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

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 0.5%

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

   9. Applied egg-rr 0.5%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 0.5%

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

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

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

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

       3. *-commutative N/A

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

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

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

       5. metadata-eval 27.4%

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

   14. Simplified 27.4%

       \[\leadsto \frac{\color{blue}{a \cdot 9 + -3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]

   if -8.49999999999999992e76 < rand < 2.3e154

   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 87.1%

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

   7. Simplified 87.1%

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

   if 2.3e154 < rand

   1. Initial program 99.5%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 6.1%

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

   7. Simplified 6.1%

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

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 36.3%

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

   9. Applied egg-rr 36.3%

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

       1. clear-num N/A

          \[\leadsto \frac{\frac{1}{9} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

       2. flip3-- N/A

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

       3. associate-/l/ N/A

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

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

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

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

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

       6. metadata-eval N/A

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

       7. cube-mult N/A

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

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

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

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

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

       10. associate-*l* N/A

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

       11. cube-mult N/A

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

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

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

       13. cube-mult N/A

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

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

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

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

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

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

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

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

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

   11. Applied egg-rr 0.6%

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

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{729}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \color{blue}{\frac{-1}{243}}\right) \]
   13. Step-by-step derivation

   14. Simplified 40.4%

       \[\leadsto \frac{0.0013717421124828531 - \left(a \cdot a\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{-0.00411522633744856}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 72.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{0.1111111111111111 - a \cdot a}\\ \mathbf{if}\;rand \leq -8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot 9 + -3}{t\_0}\\ \mathbf{elif}\;rand \leq 3.5 \cdot 10^{+141}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- 0.1111111111111111 (* a a)))))
   (if (<= rand -8.5e+76)
     (/ (+ (* a 9.0) -3.0) t_0)
     (if (<= rand 3.5e+141) (+ -0.3333333333333333 a) (/ -3.0 t_0)))))

C

double code(double a, double rand) {
	double t_0 = 1.0 / (0.1111111111111111 - (a * a));
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * 9.0) + -3.0) / t_0;
	} else if (rand <= 3.5e+141) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = -3.0 / 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 = 1.0d0 / (0.1111111111111111d0 - (a * a))
    if (rand <= (-8.5d+76)) then
        tmp = ((a * 9.0d0) + (-3.0d0)) / t_0
    else if (rand <= 3.5d+141) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (-3.0d0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 1.0 / (0.1111111111111111 - (a * a));
	double tmp;
	if (rand <= -8.5e+76) {
		tmp = ((a * 9.0) + -3.0) / t_0;
	} else if (rand <= 3.5e+141) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = -3.0 / t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 1.0 / (0.1111111111111111 - (a * a))
	tmp = 0
	if rand <= -8.5e+76:
		tmp = ((a * 9.0) + -3.0) / t_0
	elif rand <= 3.5e+141:
		tmp = -0.3333333333333333 + a
	else:
		tmp = -3.0 / t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(1.0 / Float64(0.1111111111111111 - Float64(a * a)))
	tmp = 0.0
	if (rand <= -8.5e+76)
		tmp = Float64(Float64(Float64(a * 9.0) + -3.0) / t_0);
	elseif (rand <= 3.5e+141)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(-3.0 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 1.0 / (0.1111111111111111 - (a * a));
	tmp = 0.0;
	if (rand <= -8.5e+76)
		tmp = ((a * 9.0) + -3.0) / t_0;
	elseif (rand <= 3.5e+141)
		tmp = -0.3333333333333333 + a;
	else
		tmp = -3.0 / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(1.0 / N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.5e+76], N[(N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[rand, 3.5e+141], N[(-0.3333333333333333 + a), $MachinePrecision], N[(-3.0 / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -8.49999999999999992e76

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   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 0.6%

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

   7. Simplified 0.6%

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

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

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

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 0.5%

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

   9. Applied egg-rr 0.5%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 0.5%

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

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

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

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

       3. *-commutative N/A

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

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

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

       5. metadata-eval 27.4%

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

   14. Simplified 27.4%

       \[\leadsto \frac{\color{blue}{a \cdot 9 + -3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]

   if -8.49999999999999992e76 < rand < 3.5e141

   1. Initial program 99.9%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 88.4%

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

   7. Simplified 88.4%

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

   if 3.5e141 < rand

   1. Initial program 99.5%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 6.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 6.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{3} - a}{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{3} - a}{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{3} - a\right), \color{blue}{\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right)}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \left(\color{blue}{\frac{-1}{3} \cdot \frac{-1}{3}} - a \cdot a\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \mathsf{\_.f64}\left(\frac{1}{9}, \left(\color{blue}{a} \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 33.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{-0.3333333333333333 - a}{0.1111111111111111 - a \cdot a}}} \]
   10. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \frac{1}{\left(\frac{-1}{3} - a\right) \cdot \color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{-1}{3} - a}\right), \color{blue}{\left(\frac{1}{\frac{1}{9} - a \cdot a}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{-1}{3} - a\right)\right), \left(\frac{\color{blue}{1}}{\frac{1}{9} - a \cdot a}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{\frac{1}{9} - a \cdot a}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{9} - a \cdot a\right)}\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{9}, \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 33.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   11. Applied egg-rr 33.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{-3}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 34.9%

       \[\leadsto \frac{\color{blue}{-3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 67.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 5.4 \cdot 10^{+142}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\frac{1}{0.1111111111111111 - a \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 5.4e+142)
   (+ -0.3333333333333333 a)
   (/ -3.0 (/ 1.0 (- 0.1111111111111111 (* a a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 5.4e+142) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = -3.0 / (1.0 / (0.1111111111111111 - (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 <= 5.4d+142) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (-3.0d0) / (1.0d0 / (0.1111111111111111d0 - (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 5.4e+142) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = -3.0 / (1.0 / (0.1111111111111111 - (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 5.4e+142:
		tmp = -0.3333333333333333 + a
	else:
		tmp = -3.0 / (1.0 / (0.1111111111111111 - (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 5.4e+142)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(-3.0 / Float64(1.0 / Float64(0.1111111111111111 - Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 5.4e+142)
		tmp = -0.3333333333333333 + a;
	else
		tmp = -3.0 / (1.0 / (0.1111111111111111 - (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 5.4e+142], N[(-0.3333333333333333 + a), $MachinePrecision], N[(-3.0 / N[(1.0 / N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 5.39999999999999965e142

   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 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   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 69.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 69.0%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 5.39999999999999965e142 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 6.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 6.3%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   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. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{3} - a}{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{3} - a}{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{3} - a\right), \color{blue}{\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right)}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \left(\color{blue}{\frac{-1}{3} \cdot \frac{-1}{3}} - a \cdot a\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \mathsf{\_.f64}\left(\frac{1}{9}, \left(\color{blue}{a} \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 33.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{3}, a\right), \mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{-0.3333333333333333 - a}{0.1111111111111111 - a \cdot a}}} \]
   10. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \frac{1}{\left(\frac{-1}{3} - a\right) \cdot \color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{-1}{3} - a}}{\color{blue}{\frac{1}{\frac{1}{9} - a \cdot a}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{-1}{3} - a}\right), \color{blue}{\left(\frac{1}{\frac{1}{9} - a \cdot a}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{-1}{3} - a\right)\right), \left(\frac{\color{blue}{1}}{\frac{1}{9} - a \cdot a}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{\frac{1}{9} - a \cdot a}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{9} - a \cdot a\right)}\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{9}, \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 33.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   11. Applied egg-rr 33.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333 - a}}{\frac{1}{0.1111111111111111 - a \cdot a}}} \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{-3}, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 34.9%

       \[\leadsto \frac{\color{blue}{-3}}{\frac{1}{0.1111111111111111 - a \cdot a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 63.1% 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 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
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 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

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 60.7%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 60.7%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
8. Add Preprocessing

Alternative 21: 61.9% accurate, 119.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 a)

C

double code(double a, double rand) {
	return a;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a
end function

Java

public static double code(double a, double rand) {
	return a;
}

Python

def code(a, rand):
	return a

Julia

function code(a, rand)
	return a
end

MATLAB

function tmp = code(a, rand)
	tmp = a;
end

Wolfram

code[a_, rand_] := a

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

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 60.7%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 60.7%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 59.9%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Alternative 22: 1.5% accurate, 119.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 -0.3333333333333333)

C

double code(double a, double rand) {
	return -0.3333333333333333;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = -0.3333333333333333d0
end function

Java

public static double code(double a, double rand) {
	return -0.3333333333333333;
}

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

function tmp = code(a, rand)
	tmp = -0.3333333333333333;
end

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

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 60.7%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 60.7%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{3}} \]
9. Step-by-step derivation

10. Simplified 1.7%

    \[\leadsto \color{blue}{-0.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))