Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.9%

Time: 12.6s

Alternatives: 18

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: 99.8% 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

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

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

   10. +-commutative N/A

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

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

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

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

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

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

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

   14. +-lowering-+.f64 99.9%

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

7. Applied egg-rr 99.9%

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

   1. associate-/r/ N/A

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

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

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

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

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

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

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

   5. +-lowering-+.f64 99.9%

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

9. Applied egg-rr 99.9%

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

Alternative 2: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -8.8 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2 \cdot 10^{+59}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ a -0.3333333333333333)) (* rand 0.3333333333333333))))
   (if (<= rand -8.8e+86)
     t_0
     (if (<= rand 2e+59) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -8.8e+86) {
		tmp = t_0;
	} else if (rand <= 2e+59) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a + (-0.3333333333333333d0))) * (rand * 0.3333333333333333d0)
    if (rand <= (-8.8d+86)) then
        tmp = t_0
    else if (rand <= 2d+59) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -8.8e+86) {
		tmp = t_0;
	} else if (rand <= 2e+59) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333)
	tmp = 0
	if rand <= -8.8e+86:
		tmp = t_0
	elif rand <= 2e+59:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(a + -0.3333333333333333)) * Float64(rand * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -8.8e+86)
		tmp = t_0;
	elseif (rand <= 2e+59)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -8.8e+86)
		tmp = t_0;
	elseif (rand <= 2e+59)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.8e+86], t$95$0, If[LessEqual[rand, 2e+59], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -8.80000000000000013e86 or 1.99999999999999994e59 < 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. Add Preprocessing

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      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. *-commutative N/A

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

      10. *-lowering-*.f64 93.5%

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

   5. Simplified 93.5%

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

   if -8.80000000000000013e86 < rand < 1.99999999999999994e59

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 96.8%

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

   7. Simplified 96.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8.8 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 2 \cdot 10^{+59}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -3.05 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 3.1 \cdot 10^{+60}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* (sqrt (+ a -0.3333333333333333)) 0.3333333333333333))))
   (if (<= rand -3.05e+85)
     t_0
     (if (<= rand 3.1e+60) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	double tmp;
	if (rand <= -3.05e+85) {
		tmp = t_0;
	} else if (rand <= 3.1e+60) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = rand * (sqrt((a + (-0.3333333333333333d0))) * 0.3333333333333333d0)
    if (rand <= (-3.05d+85)) then
        tmp = t_0
    else if (rand <= 3.1d+60) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand * (Math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	double tmp;
	if (rand <= -3.05e+85) {
		tmp = t_0;
	} else if (rand <= 3.1e+60) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333)
	tmp = 0
	if rand <= -3.05e+85:
		tmp = t_0
	elif rand <= 3.1e+60:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(sqrt(Float64(a + -0.3333333333333333)) * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -3.05e+85)
		tmp = t_0;
	elseif (rand <= 3.1e+60)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -3.05e+85)
		tmp = t_0;
	elseif (rand <= 3.1e+60)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -3.05e+85], t$95$0, If[LessEqual[rand, 3.1e+60], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -3.04999999999999991e85 or 3.1000000000000001e60 < 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. Add Preprocessing

   3. Taylor expanded in rand around inf

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

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

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. +-commutative N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in rand around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 93.5%

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

   8. Simplified 93.5%

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

   if -3.04999999999999991e85 < rand < 3.1000000000000001e60

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 96.8%

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

   7. Simplified 96.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -3.05 \cdot 10^{+85}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 3.1 \cdot 10^{+60}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.85 \cdot 10^{+87}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 3.1 \cdot 10^{+60}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.85e+87)
   (* (* rand 0.3333333333333333) (sqrt a))
   (if (<= rand 3.1e+60)
     (+ a -0.3333333333333333)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.85e+87) {
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	} else if (rand <= 3.1e+60) {
		tmp = a + -0.3333333333333333;
	} 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 <= (-1.85d+87)) then
        tmp = (rand * 0.3333333333333333d0) * sqrt(a)
    else if (rand <= 3.1d+60) then
        tmp = a + (-0.3333333333333333d0)
    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 <= -1.85e+87) {
		tmp = (rand * 0.3333333333333333) * Math.sqrt(a);
	} else if (rand <= 3.1e+60) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.85e+87:
		tmp = (rand * 0.3333333333333333) * math.sqrt(a)
	elif rand <= 3.1e+60:
		tmp = a + -0.3333333333333333
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.85e+87)
		tmp = Float64(Float64(rand * 0.3333333333333333) * sqrt(a));
	elseif (rand <= 3.1e+60)
		tmp = Float64(a + -0.3333333333333333);
	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 <= -1.85e+87)
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	elseif (rand <= 3.1e+60)
		tmp = a + -0.3333333333333333;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.85e+87], N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 3.1e+60], N[(a + -0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.85000000000000001e87

   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. Add Preprocessing

   3. Taylor expanded in rand around 0

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

      1. associate--l+ N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)} \]
   10. 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

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

       3. associate-*l* N/A

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

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

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

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

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

       6. *-lowering-*.f64 91.7%

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

   11. Simplified 91.7%

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

   if -1.85000000000000001e87 < rand < 3.1000000000000001e60

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 96.8%

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

   7. Simplified 96.8%

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

   if 3.1000000000000001e60 < rand

   1. Initial program 99.4%

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

   3. Taylor expanded in rand around 0

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

      1. associate--l+ N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 96.3%

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

   8. Simplified 96.3%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. associate-/r/ N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 96.3%

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

   10. Applied egg-rr 96.3%

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

   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 89.3%

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

   13. Simplified 89.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.85 \cdot 10^{+87}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 3.1 \cdot 10^{+60}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -4.5 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2.75 \cdot 10^{+60}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -4.5e+87)
     t_0
     (if (<= rand 2.75e+60) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -4.5e+87) {
		tmp = t_0;
	} else if (rand <= 2.75e+60) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (rand * sqrt(a))
    if (rand <= (-4.5d+87)) then
        tmp = t_0
    else if (rand <= 2.75d+60) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * Math.sqrt(a));
	double tmp;
	if (rand <= -4.5e+87) {
		tmp = t_0;
	} else if (rand <= 2.75e+60) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.3333333333333333 * (rand * math.sqrt(a))
	tmp = 0
	if rand <= -4.5e+87:
		tmp = t_0
	elif rand <= 2.75e+60:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.3333333333333333 * Float64(rand * sqrt(a)))
	tmp = 0.0
	if (rand <= -4.5e+87)
		tmp = t_0;
	elseif (rand <= 2.75e+60)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.3333333333333333 * (rand * sqrt(a));
	tmp = 0.0;
	if (rand <= -4.5e+87)
		tmp = t_0;
	elseif (rand <= 2.75e+60)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -4.5e+87], t$95$0, If[LessEqual[rand, 2.75e+60], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -4.5000000000000003e87 or 2.75e60 < 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. Add Preprocessing

   3. Taylor expanded in rand around 0

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

      1. associate--l+ N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 96.5%

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

   8. Simplified 96.5%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. associate-/r/ N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 96.5%

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

   10. Applied egg-rr 96.5%

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

   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 90.4%

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

   13. Simplified 90.4%

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

   if -4.5000000000000003e87 < rand < 2.75e60

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 96.8%

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

   7. Simplified 96.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.5 \cdot 10^{+87}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 2.75 \cdot 10^{+60}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-*r/ N/A

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

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

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

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

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

   6. +-commutative N/A

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

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

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

   8. +-lowering-+.f64 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 7: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 8: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ a + \frac{\sqrt{a + -0.3333333333333333}}{\frac{3}{rand}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

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

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

   10. +-commutative N/A

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

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

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

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

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

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

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

   14. +-lowering-+.f64 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in a around inf

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

10. Simplified 98.7%

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

11. Final simplification 98.7%

    \[\leadsto a + \frac{\sqrt{a + -0.3333333333333333}}{\frac{3}{rand}} \]
12. Add Preprocessing

Alternative 9: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in rand around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. *-lowering-*.f64 98.6%

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

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

Alternative 10: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in a around inf

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

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

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

   2. *-commutative N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 97.2%

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

8. Simplified 97.2%

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. associate-/r/ N/A

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

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

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

   5. clear-num N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 97.2%

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

10. Applied egg-rr 97.2%

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

    1. associate-*l/ N/A

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

    2. associate-/l* N/A

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

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

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

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

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

    5. sqrt-lowering-sqrt.f64 97.2%

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

12. Applied egg-rr 97.2%

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

Alternative 11: 98.0% 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. Add Preprocessing

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in a around inf

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

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

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

   2. *-commutative N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 97.2%

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

8. Simplified 97.2%

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

Alternative 12: 75.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\frac{rand}{a \cdot a - 0.1111111111111111}}\\ \mathbf{if}\;rand \leq -3.5 \cdot 10^{+123}:\\ \;\;\;\;t\_0 \cdot \left(rand \cdot \left(a \cdot 9 + -3\right)\right)\\ \mathbf{elif}\;rand \leq 2.6 \cdot 10^{+152}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(rand \cdot -3 + a \cdot \left(rand \cdot \left(9 + a \cdot -27\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ -1.0 (/ rand (- (* a a) 0.1111111111111111)))))
   (if (<= rand -3.5e+123)
     (* t_0 (* rand (+ (* a 9.0) -3.0)))
     (if (<= rand 2.6e+152)
       (+ a -0.3333333333333333)
       (* t_0 (+ (* rand -3.0) (* a (* rand (+ 9.0 (* a -27.0))))))))))

C

double code(double a, double rand) {
	double t_0 = -1.0 / (rand / ((a * a) - 0.1111111111111111));
	double tmp;
	if (rand <= -3.5e+123) {
		tmp = t_0 * (rand * ((a * 9.0) + -3.0));
	} else if (rand <= 2.6e+152) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * ((rand * -3.0) + (a * (rand * (9.0 + (a * -27.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / (rand / ((a * a) - 0.1111111111111111d0))
    if (rand <= (-3.5d+123)) then
        tmp = t_0 * (rand * ((a * 9.0d0) + (-3.0d0)))
    else if (rand <= 2.6d+152) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * ((rand * (-3.0d0)) + (a * (rand * (9.0d0 + (a * (-27.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = -1.0 / (rand / ((a * a) - 0.1111111111111111));
	double tmp;
	if (rand <= -3.5e+123) {
		tmp = t_0 * (rand * ((a * 9.0) + -3.0));
	} else if (rand <= 2.6e+152) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * ((rand * -3.0) + (a * (rand * (9.0 + (a * -27.0)))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = -1.0 / (rand / ((a * a) - 0.1111111111111111))
	tmp = 0
	if rand <= -3.5e+123:
		tmp = t_0 * (rand * ((a * 9.0) + -3.0))
	elif rand <= 2.6e+152:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * ((rand * -3.0) + (a * (rand * (9.0 + (a * -27.0)))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(-1.0 / Float64(rand / Float64(Float64(a * a) - 0.1111111111111111)))
	tmp = 0.0
	if (rand <= -3.5e+123)
		tmp = Float64(t_0 * Float64(rand * Float64(Float64(a * 9.0) + -3.0)));
	elseif (rand <= 2.6e+152)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(Float64(rand * -3.0) + Float64(a * Float64(rand * Float64(9.0 + Float64(a * -27.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = -1.0 / (rand / ((a * a) - 0.1111111111111111));
	tmp = 0.0;
	if (rand <= -3.5e+123)
		tmp = t_0 * (rand * ((a * 9.0) + -3.0));
	elseif (rand <= 2.6e+152)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * ((rand * -3.0) + (a * (rand * (9.0 + (a * -27.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(-1.0 / N[(rand / N[(N[(a * a), $MachinePrecision] - 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -3.5e+123], N[(t$95$0 * N[(rand * N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 2.6e+152], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(N[(rand * -3.0), $MachinePrecision] + N[(a * N[(rand * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -3.5e123

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

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

   7. Simplified 0.4%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\color{blue}{0}} \]

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. inv-pow N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. associate-*l/ N/A

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

      12. flip-+ N/A

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

      13. associate-/r/ N/A

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

      14. times-frac N/A

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

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

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

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

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

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

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

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

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

      19. metadata-eval N/A

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

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

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

      21. /-lowering-/.f64 N/A

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

      22. --lowering--.f64 0.3%

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

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-out N/A

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

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

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 52.2%

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

   12. Simplified 52.2%

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

   if -3.5e123 < rand < 2.6000000000000001e152

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 91.7%

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

   7. Simplified 91.7%

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

   if 2.6000000000000001e152 < rand

   1. Initial program 99.4%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.6%

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

   3. Simplified 99.6%

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

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\color{blue}{0}} \]

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. inv-pow N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. associate-*l/ N/A

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

      12. flip-+ N/A

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

      13. associate-/r/ N/A

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

      14. times-frac N/A

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

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

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

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

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

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

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

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

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

      19. metadata-eval N/A

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

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

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

      21. /-lowering-/.f64 N/A

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

      22. --lowering--.f64 40.2%

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

   9. Applied egg-rr 40.2%

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

   10. Taylor expanded in a around 0

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

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

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

       2. *-commutative N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 51.3%

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

   12. Simplified 51.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -3.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{\frac{rand}{a \cdot a - 0.1111111111111111}} \cdot \left(rand \cdot \left(a \cdot 9 + -3\right)\right)\\ \mathbf{elif}\;rand \leq 2.6 \cdot 10^{+152}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{rand}{a \cdot a - 0.1111111111111111}} \cdot \left(rand \cdot -3 + a \cdot \left(rand \cdot \left(9 + a \cdot -27\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -3.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{\frac{rand}{a \cdot a - 0.1111111111111111}} \cdot \left(rand \cdot \left(a \cdot 9 + -3\right)\right)\\ \mathbf{elif}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -3.5e+123)
   (*
    (/ -1.0 (/ rand (- (* a a) 0.1111111111111111)))
    (* rand (+ (* a 9.0) -3.0)))
   (if (<= rand 4.1e+154) (+ a -0.3333333333333333) (/ (* a rand) rand))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -3.5e+123) {
		tmp = (-1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand * ((a * 9.0) + -3.0));
	} else if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-3.5d+123)) then
        tmp = ((-1.0d0) / (rand / ((a * a) - 0.1111111111111111d0))) * (rand * ((a * 9.0d0) + (-3.0d0)))
    else if (rand <= 4.1d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (a * rand) / rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -3.5e+123) {
		tmp = (-1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand * ((a * 9.0) + -3.0));
	} else if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -3.5e+123:
		tmp = (-1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand * ((a * 9.0) + -3.0))
	elif rand <= 4.1e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = (a * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -3.5e+123)
		tmp = Float64(Float64(-1.0 / Float64(rand / Float64(Float64(a * a) - 0.1111111111111111))) * Float64(rand * Float64(Float64(a * 9.0) + -3.0)));
	elseif (rand <= 4.1e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(a * rand) / rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -3.5e+123)
		tmp = (-1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand * ((a * 9.0) + -3.0));
	elseif (rand <= 4.1e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = (a * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -3.5e+123], N[(N[(-1.0 / N[(rand / N[(N[(a * a), $MachinePrecision] - 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(rand * N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.1e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(a * rand), $MachinePrecision] / rand), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -3.5e123

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

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

   7. Simplified 0.4%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\color{blue}{0}} \]

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. inv-pow N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. associate-*l/ N/A

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

      12. flip-+ N/A

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

      13. associate-/r/ N/A

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

      14. times-frac N/A

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

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

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

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

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

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

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

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

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

      19. metadata-eval N/A

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

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

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

      21. /-lowering-/.f64 N/A

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

      22. --lowering--.f64 0.3%

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

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-out N/A

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

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

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 52.2%

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

   12. Simplified 52.2%

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

   if -3.5e123 < rand < 4.1e154

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 91.7%

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

   7. Simplified 91.7%

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

   if 4.1e154 < rand

   1. Initial program 99.4%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.6%

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

   3. Simplified 99.6%

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

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\color{blue}{0}} \]

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. inv-pow N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. div-inv N/A

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

      10. +-commutative N/A

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

      11. associate-*l/ N/A

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

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

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

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

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

      14. +-lowering-+.f64 51.0%

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

   9. Applied egg-rr 51.0%

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

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot rand\right)}, rand\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-lowering-*.f64 51.0%

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

   12. Simplified 51.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -3.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{\frac{rand}{a \cdot a - 0.1111111111111111}} \cdot \left(rand \cdot \left(a \cdot 9 + -3\right)\right)\\ \mathbf{elif}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -6.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{rand}{-0.3333333333333333 - a}}{\frac{rand}{0.1111111111111111 - a \cdot a}}\\ \mathbf{elif}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -6.8e+127)
   (/
    (/ rand (- -0.3333333333333333 a))
    (/ rand (- 0.1111111111111111 (* a a))))
   (if (<= rand 4.1e+154) (+ a -0.3333333333333333) (/ (* a rand) rand))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -6.8e+127) {
		tmp = (rand / (-0.3333333333333333 - a)) / (rand / (0.1111111111111111 - (a * a)));
	} else if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-6.8d+127)) then
        tmp = (rand / ((-0.3333333333333333d0) - a)) / (rand / (0.1111111111111111d0 - (a * a)))
    else if (rand <= 4.1d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (a * rand) / rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -6.8e+127) {
		tmp = (rand / (-0.3333333333333333 - a)) / (rand / (0.1111111111111111 - (a * a)));
	} else if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -6.8e+127:
		tmp = (rand / (-0.3333333333333333 - a)) / (rand / (0.1111111111111111 - (a * a)))
	elif rand <= 4.1e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = (a * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -6.8e+127)
		tmp = Float64(Float64(rand / Float64(-0.3333333333333333 - a)) / Float64(rand / Float64(0.1111111111111111 - Float64(a * a))));
	elseif (rand <= 4.1e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(a * rand) / rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -6.8e+127)
		tmp = (rand / (-0.3333333333333333 - a)) / (rand / (0.1111111111111111 - (a * a)));
	elseif (rand <= 4.1e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = (a * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -6.8e+127], N[(N[(rand / N[(-0.3333333333333333 - a), $MachinePrecision]), $MachinePrecision] / N[(rand / N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.1e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(a * rand), $MachinePrecision] / rand), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -6.79999999999999955e127

   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. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

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

   7. Simplified 0.4%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\color{blue}{0}} \]

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. inv-pow N/A

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

      7. associate-*l* N/A

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

      8. +-commutative N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

      14. +-commutative N/A

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

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

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

      16. +-lowering-+.f64 0.4%

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

   9. Applied egg-rr 0.4%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

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

       3. *-inverses N/A

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

       4. flip-+ N/A

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

       5. clear-num N/A

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

       6. clear-num N/A

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

       7. flip-+ N/A

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

       8. +-commutative N/A

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

       9. flip-+ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

   11. Applied egg-rr 42.3%

       \[\leadsto \color{blue}{\frac{-\frac{rand}{-0.3333333333333333 - a}}{0 - \frac{rand}{0.1111111111111111 - a \cdot a}}} \]

   if -6.79999999999999955e127 < rand < 4.1e154

   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. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a\right)\right)\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 9 \cdot a\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a \cdot 9\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \frac{-1}{3}\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, 9\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 91.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 91.2%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 4.1e154 < rand

   1. Initial program 99.4%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a\right)\right)\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 9 \cdot a\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a \cdot 9\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \frac{-1}{3}\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, 9\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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. +-commutative N/A

         \[\leadsto a + \color{blue}{\frac{-1}{3}} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \color{blue}{1} \]

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\color{blue}{0}} \]

      4. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\left(-1 + \color{blue}{1}\right)} \]

      5. pow-plus N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \left({rand}^{-1} \cdot \color{blue}{rand}\right) \]

      6. inv-pow N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \left(\frac{1}{rand} \cdot rand\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(a + \frac{-1}{3}\right) \cdot \frac{1}{rand}\right) \cdot \color{blue}{rand} \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{3} + a\right) \cdot \frac{1}{rand}\right) \cdot rand \]

      9. div-inv N/A

         \[\leadsto \frac{\frac{-1}{3} + a}{rand} \cdot rand \]

      10. +-commutative N/A

          \[\leadsto \frac{a + \frac{-1}{3}}{rand} \cdot rand \]

      11. associate-*l/ N/A

          \[\leadsto \frac{\left(a + \frac{-1}{3}\right) \cdot rand}{\color{blue}{rand}} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(a + \frac{-1}{3}\right) \cdot rand\right), \color{blue}{rand}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{3}\right), rand\right), rand\right) \]

      14. +-lowering-+.f64 51.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), rand\right), rand\right) \]

   9. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\frac{\left(a + -0.3333333333333333\right) \cdot rand}{rand}} \]

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot rand\right)}, rand\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(rand \cdot a\right), rand\right) \]

       2. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(rand, a\right), rand\right) \]

   12. Simplified 51.0%

       \[\leadsto \frac{\color{blue}{rand \cdot a}}{rand} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -6.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{rand}{-0.3333333333333333 - a}}{\frac{rand}{0.1111111111111111 - a \cdot a}}\\ \mathbf{elif}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 4.1e+154) (+ a -0.3333333333333333) (/ (* a rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 4.1d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (a * rand) / rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 4.1e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = (a * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 4.1e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(a * rand) / rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 4.1e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = (a * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 4.1e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(a * rand), $MachinePrecision] / rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 4.1e154

   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. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a\right)\right)\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 9 \cdot a\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a \cdot 9\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \frac{-1}{3}\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, 9\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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 77.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 77.2%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 4.1e154 < rand

   1. Initial program 99.4%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a\right)\right)\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 9 \cdot a\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a \cdot 9\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \frac{-1}{3}\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, 9\right)\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
   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. +-commutative N/A

         \[\leadsto a + \color{blue}{\frac{-1}{3}} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \color{blue}{1} \]

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\color{blue}{0}} \]

      4. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot {rand}^{\left(-1 + \color{blue}{1}\right)} \]

      5. pow-plus N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \left({rand}^{-1} \cdot \color{blue}{rand}\right) \]

      6. inv-pow N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \left(\frac{1}{rand} \cdot rand\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(a + \frac{-1}{3}\right) \cdot \frac{1}{rand}\right) \cdot \color{blue}{rand} \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{3} + a\right) \cdot \frac{1}{rand}\right) \cdot rand \]

      9. div-inv N/A

         \[\leadsto \frac{\frac{-1}{3} + a}{rand} \cdot rand \]

      10. +-commutative N/A

          \[\leadsto \frac{a + \frac{-1}{3}}{rand} \cdot rand \]

      11. associate-*l/ N/A

          \[\leadsto \frac{\left(a + \frac{-1}{3}\right) \cdot rand}{\color{blue}{rand}} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(a + \frac{-1}{3}\right) \cdot rand\right), \color{blue}{rand}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a + \frac{-1}{3}\right), rand\right), rand\right) \]

      14. +-lowering-+.f64 51.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), rand\right), rand\right) \]

   9. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\frac{\left(a + -0.3333333333333333\right) \cdot rand}{rand}} \]

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(a \cdot rand\right)}, rand\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(rand \cdot a\right), rand\right) \]

       2. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(rand, a\right), rand\right) \]

   12. Simplified 51.0%

       \[\leadsto \frac{\color{blue}{rand \cdot a}}{rand} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.4% accurate, 39.7× speedup

Math

\[\begin{array}{l} \\ a + -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 (+ a -0.3333333333333333))

C

double code(double a, double rand) {
	return a + -0.3333333333333333;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a + (-0.3333333333333333d0)
end function

Java

public static double code(double a, double rand) {
	return a + -0.3333333333333333;
}

Python

def code(a, rand):
	return a + -0.3333333333333333

Julia

function code(a, rand)
	return Float64(a + -0.3333333333333333)
end

MATLAB

function tmp = code(a, rand)
	tmp = a + -0.3333333333333333;
end

Wolfram

code[a_, rand_] := N[(a + -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a\right)\right)\right)\right)\right)\right) \]

   13. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 9 \cdot a\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a \cdot 9\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \frac{-1}{3}\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, 9\right)\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
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 67.2%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 67.2%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Final simplification 67.2%

   \[\leadsto a + -0.3333333333333333 \]
9. Add Preprocessing

Alternative 17: 62.5% 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. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a\right)\right)\right)\right)\right)\right) \]

   13. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 9 \cdot a\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a \cdot 9\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \frac{-1}{3}\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, 9\right)\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
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 67.2%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 67.2%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 66.0%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Alternative 18: 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. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a\right)\right)\right)\right)\right)\right) \]

   13. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 9 \cdot a\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + a \cdot 9\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(9 \cdot \frac{-1}{3}\right), \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \left(a \cdot 9\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(-3, \mathsf{*.f64}\left(a, 9\right)\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{-3 + a \cdot 9}}\right)} \]
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 67.2%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 67.2%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{3}} \]
9. Step-by-step derivation

10. Simplified 1.5%

    \[\leadsto \color{blue}{-0.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))))