Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 11.3s

Alternatives: 21

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

7. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

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

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

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

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. pow1/2 N/A

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

   11. +-commutative N/A

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

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

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

   13. +-lowering-+.f64 99.8%

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

9. Applied egg-rr 99.8%

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

10. Taylor expanded in rand around 0

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

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

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. +-lowering-+.f64 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{if}\;rand \leq -1 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 7.5 \cdot 10^{+98}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ a -0.3333333333333333)) (* 0.3333333333333333 rand))))
   (if (<= rand -1e+78)
     t_0
     (if (<= rand 7.5e+98) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333)) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -1e+78) {
		tmp = t_0;
	} else if (rand <= 7.5e+98) {
		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))) * (0.3333333333333333d0 * rand)
    if (rand <= (-1d+78)) then
        tmp = t_0
    else if (rand <= 7.5d+98) 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)) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -1e+78) {
		tmp = t_0;
	} else if (rand <= 7.5e+98) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333)) * (0.3333333333333333 * rand)
	tmp = 0
	if rand <= -1e+78:
		tmp = t_0
	elif rand <= 7.5e+98:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(a + -0.3333333333333333)) * Float64(0.3333333333333333 * rand))
	tmp = 0.0
	if (rand <= -1e+78)
		tmp = t_0;
	elseif (rand <= 7.5e+98)
		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)) * (0.3333333333333333 * rand);
	tmp = 0.0;
	if (rand <= -1e+78)
		tmp = t_0;
	elseif (rand <= 7.5e+98)
		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[(0.3333333333333333 * rand), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1e+78], t$95$0, If[LessEqual[rand, 7.5e+98], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.00000000000000001e78 or 7.50000000000000036e98 < rand

   1. Initial program 99.5%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 89.3%

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

   7. Simplified 89.3%

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

   if -1.00000000000000001e78 < rand < 7.50000000000000036e98

   1. Initial program 100.0%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 94.4%

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

   7. Simplified 94.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{elif}\;rand \leq 7.5 \cdot 10^{+98}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.5 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{a} \cdot \frac{rand}{3}\\ \mathbf{elif}\;rand \leq 1.8 \cdot 10^{+99}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{a}}{\frac{3}{rand}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.5e+76)
   (* (sqrt a) (/ rand 3.0))
   (if (<= rand 1.8e+99) (+ a -0.3333333333333333) (/ (sqrt a) (/ 3.0 rand)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.5e+76) {
		tmp = sqrt(a) * (rand / 3.0);
	} else if (rand <= 1.8e+99) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = sqrt(a) / (3.0 / 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 <= (-1.5d+76)) then
        tmp = sqrt(a) * (rand / 3.0d0)
    else if (rand <= 1.8d+99) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = sqrt(a) / (3.0d0 / rand)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.5e+76) {
		tmp = Math.sqrt(a) * (rand / 3.0);
	} else if (rand <= 1.8e+99) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = Math.sqrt(a) / (3.0 / rand);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.5e+76:
		tmp = math.sqrt(a) * (rand / 3.0)
	elif rand <= 1.8e+99:
		tmp = a + -0.3333333333333333
	else:
		tmp = math.sqrt(a) / (3.0 / rand)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.5e+76)
		tmp = Float64(sqrt(a) * Float64(rand / 3.0));
	elseif (rand <= 1.8e+99)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(sqrt(a) / Float64(3.0 / rand));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.5e+76)
		tmp = sqrt(a) * (rand / 3.0);
	elseif (rand <= 1.8e+99)
		tmp = a + -0.3333333333333333;
	else
		tmp = sqrt(a) / (3.0 / rand);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.5e+76], N[(N[Sqrt[a], $MachinePrecision] * N[(rand / 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.8e+99], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[Sqrt[a], $MachinePrecision] / N[(3.0 / rand), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.4999999999999999e76

   1. Initial program 99.5%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 85.3%

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

   10. Simplified 85.3%

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. div-inv N/A

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

       4. associate-*l/ N/A

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 85.5%

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

   12. Applied egg-rr 85.5%

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

   if -1.4999999999999999e76 < rand < 1.8000000000000001e99

   1. Initial program 100.0%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 94.4%

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

   7. Simplified 94.4%

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

   if 1.8000000000000001e99 < 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. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 91.8%

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

   10. Simplified 91.8%

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. associate-/r/ N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

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

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

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

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

       8. /-lowering-/.f64 92.0%

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

   12. Applied egg-rr 92.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.5 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{a} \cdot \frac{rand}{3}\\ \mathbf{elif}\;rand \leq 1.8 \cdot 10^{+99}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{a}}{\frac{3}{rand}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a} \cdot \frac{rand}{3}\\ \mathbf{if}\;rand \leq -5.6 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt a) (/ rand 3.0))))
   (if (<= rand -5.6e+78)
     t_0
     (if (<= rand 9.5e+98) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt(a) * (rand / 3.0);
	double tmp;
	if (rand <= -5.6e+78) {
		tmp = t_0;
	} else if (rand <= 9.5e+98) {
		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) * (rand / 3.0d0)
    if (rand <= (-5.6d+78)) then
        tmp = t_0
    else if (rand <= 9.5d+98) 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) * (rand / 3.0);
	double tmp;
	if (rand <= -5.6e+78) {
		tmp = t_0;
	} else if (rand <= 9.5e+98) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt(a) * (rand / 3.0)
	tmp = 0
	if rand <= -5.6e+78:
		tmp = t_0
	elif rand <= 9.5e+98:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(a) * Float64(rand / 3.0))
	tmp = 0.0
	if (rand <= -5.6e+78)
		tmp = t_0;
	elseif (rand <= 9.5e+98)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt(a) * (rand / 3.0);
	tmp = 0.0;
	if (rand <= -5.6e+78)
		tmp = t_0;
	elseif (rand <= 9.5e+98)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[a], $MachinePrecision] * N[(rand / 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -5.6e+78], t$95$0, If[LessEqual[rand, 9.5e+98], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -5.6000000000000002e78 or 9.5000000000000001e98 < rand

   1. Initial program 99.5%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 88.2%

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

   10. Simplified 88.2%

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. div-inv N/A

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

       4. associate-*l/ N/A

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 88.3%

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

   12. Applied egg-rr 88.3%

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

   if -5.6000000000000002e78 < rand < 9.5000000000000001e98

   1. Initial program 100.0%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 94.4%

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

   7. Simplified 94.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -5.6 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{a} \cdot \frac{rand}{3}\\ \mathbf{elif}\;rand \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a} \cdot \frac{rand}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.25 \cdot 10^{+80}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \frac{\sqrt{a}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.25e+80)
   (* 0.3333333333333333 (* rand (sqrt a)))
   (if (<= rand 1.15e+99)
     (+ a -0.3333333333333333)
     (* rand (/ (sqrt a) 3.0)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.25e+80) {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	} else if (rand <= 1.15e+99) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = rand * (sqrt(a) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.25d+80)) then
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    else if (rand <= 1.15d+99) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = rand * (sqrt(a) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.25e+80) {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	} else if (rand <= 1.15e+99) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = rand * (Math.sqrt(a) / 3.0);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.25e+80:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	elif rand <= 1.15e+99:
		tmp = a + -0.3333333333333333
	else:
		tmp = rand * (math.sqrt(a) / 3.0)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.25e+80)
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	elseif (rand <= 1.15e+99)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(rand * Float64(sqrt(a) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.25e+80)
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	elseif (rand <= 1.15e+99)
		tmp = a + -0.3333333333333333;
	else
		tmp = rand * (sqrt(a) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.25e+80], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.15e+99], N[(a + -0.3333333333333333), $MachinePrecision], N[(rand * N[(N[Sqrt[a], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.2499999999999999e80

   1. Initial program 99.5%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 85.3%

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

   10. Simplified 85.3%

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

   if -1.2499999999999999e80 < rand < 1.1500000000000001e99

   1. Initial program 100.0%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 94.4%

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

   7. Simplified 94.4%

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

   if 1.1500000000000001e99 < 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. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 91.8%

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

   10. Simplified 91.8%

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. div-inv N/A

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

       4. associate-/l* N/A

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 91.8%

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

   12. Applied egg-rr 91.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.25 \cdot 10^{+80}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \frac{\sqrt{a}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -2.5 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 4.5 \cdot 10^{+99}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -2.5e+80)
     t_0
     (if (<= rand 4.5e+99) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -2.5e+80) {
		tmp = t_0;
	} else if (rand <= 4.5e+99) {
		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 <= (-2.5d+80)) then
        tmp = t_0
    else if (rand <= 4.5d+99) 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 <= -2.5e+80) {
		tmp = t_0;
	} else if (rand <= 4.5e+99) {
		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 <= -2.5e+80:
		tmp = t_0
	elif rand <= 4.5e+99:
		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 <= -2.5e+80)
		tmp = t_0;
	elseif (rand <= 4.5e+99)
		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 <= -2.5e+80)
		tmp = t_0;
	elseif (rand <= 4.5e+99)
		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, -2.5e+80], t$95$0, If[LessEqual[rand, 4.5e+99], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.4999999999999998e80 or 4.5e99 < rand

   1. Initial program 99.5%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 88.2%

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

   10. Simplified 88.2%

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

   if -2.4999999999999998e80 < rand < 4.5e99

   1. Initial program 100.0%

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

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

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

      2. sub-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 94.4%

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

   7. Simplified 94.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.5 \cdot 10^{+80}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 4.5 \cdot 10^{+99}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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))) * (0.3333333333333333d0 * rand))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 8: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in a around inf

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

7. Simplified 99.3%

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

Alternative 9: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double a, double rand) {
	return (a + -0.3333333333333333) + ((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)) + ((0.3333333333333333d0 * rand) * sqrt(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

7. Simplified 99.8%

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

8. Taylor expanded in a around inf

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

   1. sqrt-lowering-sqrt.f64 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 10: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. sub-neg N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in a around inf

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

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

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

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 97.6%

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

7. Simplified 97.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

9. Applied egg-rr 97.7%

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

10. Final simplification 97.7%

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

Alternative 11: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{3} \cdot \sqrt{\frac{1}{a}}\right) \cdot \color{blue}{rand}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{1}{3}\right) \cdot rand\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right)\right) \]

   9. *-lowering-*.f64 97.6%

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right)\right) \]

7. Simplified 97.6%

   \[\leadsto \color{blue}{a \cdot \left(1 + \sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto a \cdot \left(\sqrt{\frac{1}{a}} \cdot \left(\frac{1}{3} \cdot rand\right) + \color{blue}{1}\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\sqrt{\frac{1}{a}} \cdot \left(\frac{1}{3} \cdot rand\right)\right) \cdot a + \color{blue}{1 \cdot a} \]

   3. *-lft-identity N/A

      \[\leadsto \left(\sqrt{\frac{1}{a}} \cdot \left(\frac{1}{3} \cdot rand\right)\right) \cdot a + a \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\sqrt{\frac{1}{a}} \cdot \left(\frac{1}{3} \cdot rand\right)\right) \cdot a\right), \color{blue}{a}\right) \]

9. Applied egg-rr 97.7%

   \[\leadsto \color{blue}{\frac{rand}{3} \cdot \sqrt{a} + a} \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\frac{rand}{3} \cdot \sqrt{a}\right), \color{blue}{a}\right) \]

    2. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\frac{rand \cdot \sqrt{a}}{3}\right), a\right) \]

    3. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\left(rand \cdot \frac{\sqrt{a}}{3}\right), a\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(rand, \left(\frac{\sqrt{a}}{3}\right)\right), a\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(rand, \mathsf{/.f64}\left(\left(\sqrt{a}\right), 3\right)\right), a\right) \]

    6. sqrt-lowering-sqrt.f64 97.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(rand, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(a\right), 3\right)\right), a\right) \]

11. Applied egg-rr 97.7%

    \[\leadsto \color{blue}{rand \cdot \frac{\sqrt{a}}{3} + a} \]

12. Final simplification 97.7%

    \[\leadsto a + rand \cdot \frac{\sqrt{a}}{3} \]
13. Add Preprocessing

Alternative 12: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ a + 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (+ a (* 0.3333333333333333 (* rand (sqrt a)))))

C

double code(double a, double rand) {
	return a + (0.3333333333333333 * (rand * sqrt(a)));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a + (0.3333333333333333d0 * (rand * sqrt(a)))
end function

Java

public static double code(double a, double rand) {
	return a + (0.3333333333333333 * (rand * Math.sqrt(a)));
}

Python

def code(a, rand):
	return a + (0.3333333333333333 * (rand * math.sqrt(a)))

Julia

function code(a, rand)
	return Float64(a + Float64(0.3333333333333333 * Float64(rand * sqrt(a))))
end

MATLAB

function tmp = code(a, rand)
	tmp = a + (0.3333333333333333 * (rand * sqrt(a)));
end

Wolfram

code[a_, rand_] := N[(a + N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{3} \cdot \sqrt{\frac{1}{a}}\right) \cdot \color{blue}{rand}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{1}{3}\right) \cdot rand\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right)\right) \]

   9. *-lowering-*.f64 97.6%

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right)\right) \]

7. Simplified 97.6%

   \[\leadsto \color{blue}{a \cdot \left(1 + \sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right)\right)} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{a + \frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{a} \cdot rand\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \left(rand \cdot \color{blue}{\sqrt{a}}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \color{blue}{\left(\sqrt{a}\right)}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 97.7%

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \mathsf{sqrt.f64}\left(a\right)\right)\right)\right) \]

10. Simplified 97.7%

    \[\leadsto \color{blue}{a + 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)} \]
11. Add Preprocessing

Alternative 13: 74.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3\right)\\ \mathbf{elif}\;rand \leq 5.7 \cdot 10^{+149}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -6.5e+110)
   (*
    (- 0.1111111111111111 (* a a))
    (+ (* a (+ 9.0 (* a (+ -27.0 (* a 81.0))))) -3.0))
   (if (<= rand 5.7e+149)
     (+ a -0.3333333333333333)
     (* 27.0 (* a (* a (* a a)))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	} else if (rand <= 5.7e+149) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-6.5d+110)) then
        tmp = (0.1111111111111111d0 - (a * a)) * ((a * (9.0d0 + (a * ((-27.0d0) + (a * 81.0d0))))) + (-3.0d0))
    else if (rand <= 5.7d+149) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 27.0d0 * (a * (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	} else if (rand <= 5.7e+149) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -6.5e+110:
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0)
	elif rand <= 5.7e+149:
		tmp = a + -0.3333333333333333
	else:
		tmp = 27.0 * (a * (a * (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -6.5e+110)
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(Float64(a * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * 81.0))))) + -3.0));
	elseif (rand <= 5.7e+149)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(27.0 * Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -6.5e+110)
		tmp = (0.1111111111111111 - (a * a)) * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	elseif (rand <= 5.7e+149)
		tmp = a + -0.3333333333333333;
	else
		tmp = 27.0 * (a * (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -6.5e+110], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(9.0 + N[(a * N[(-27.0 + N[(a * 81.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 5.7e+149], N[(a + -0.3333333333333333), $MachinePrecision], N[(27.0 * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -6.4999999999999997e110

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 1.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 1.0%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + a \cdot \left(81 \cdot a - 27\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot \left(81 \cdot a - 27\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a - 27\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + \left(\mathsf{neg}\left(27\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(81 \cdot a + -27\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \left(-27 + 81 \cdot a\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(81 \cdot a\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \left(a \cdot 81\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, 81\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       12. metadata-eval 44.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(-27, \mathsf{*.f64}\left(a, 81\right)\right)\right)\right)\right), -3\right)\right) \]

   12. Simplified 44.3%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3\right)} \]

   if -6.4999999999999997e110 < rand < 5.69999999999999965e149

   1. Initial program 99.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 88.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 88.1%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 5.69999999999999965e149 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 38.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 38.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]

   13. Taylor expanded in a around inf

       \[\leadsto \color{blue}{27 \cdot {a}^{4}} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \color{blue}{\left({a}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot {a}^{\color{blue}{3}}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 38.5%

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   15. Simplified 38.5%

       \[\leadsto \color{blue}{27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3\right)\\ \mathbf{elif}\;rand \leq 5.7 \cdot 10^{+149}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;-0.3333333333333333 + a \cdot \left(1 + \left(a \cdot a\right) \cdot -9\right)\\ \mathbf{elif}\;rand \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -6.5e+110)
   (+ -0.3333333333333333 (* a (+ 1.0 (* (* a a) -9.0))))
   (if (<= rand 3.4e+146)
     (+ a -0.3333333333333333)
     (* 27.0 (* a (* a (* a a)))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = -0.3333333333333333 + (a * (1.0 + ((a * a) * -9.0)));
	} else if (rand <= 3.4e+146) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-6.5d+110)) then
        tmp = (-0.3333333333333333d0) + (a * (1.0d0 + ((a * a) * (-9.0d0))))
    else if (rand <= 3.4d+146) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 27.0d0 * (a * (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = -0.3333333333333333 + (a * (1.0 + ((a * a) * -9.0)));
	} else if (rand <= 3.4e+146) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -6.5e+110:
		tmp = -0.3333333333333333 + (a * (1.0 + ((a * a) * -9.0)))
	elif rand <= 3.4e+146:
		tmp = a + -0.3333333333333333
	else:
		tmp = 27.0 * (a * (a * (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -6.5e+110)
		tmp = Float64(-0.3333333333333333 + Float64(a * Float64(1.0 + Float64(Float64(a * a) * -9.0))));
	elseif (rand <= 3.4e+146)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(27.0 * Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -6.5e+110)
		tmp = -0.3333333333333333 + (a * (1.0 + ((a * a) * -9.0)));
	elseif (rand <= 3.4e+146)
		tmp = a + -0.3333333333333333;
	else
		tmp = 27.0 * (a * (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -6.5e+110], N[(-0.3333333333333333 + N[(a * N[(1.0 + N[(N[(a * a), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 3.4e+146], N[(a + -0.3333333333333333), $MachinePrecision], N[(27.0 * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -6.4999999999999997e110

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 1.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 1.0%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 0.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 0.4%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]

   13. Taylor expanded in a around 0

       \[\leadsto \color{blue}{a \cdot \left(1 + -9 \cdot {a}^{2}\right) - \frac{1}{3}} \]
   14. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto a \cdot \left(1 + -9 \cdot {a}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto a \cdot \left(1 + -9 \cdot {a}^{2}\right) + \frac{-1}{3} \]

       3. +-commutative N/A

          \[\leadsto \frac{-1}{3} + \color{blue}{a \cdot \left(1 + -9 \cdot {a}^{2}\right)} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(a \cdot \left(1 + -9 \cdot {a}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(a, \color{blue}{\left(1 + -9 \cdot {a}^{2}\right)}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(-9 \cdot {a}^{2}\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-9, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-9, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 37.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right) \]

   15. Simplified 37.7%

       \[\leadsto \color{blue}{-0.3333333333333333 + a \cdot \left(1 + -9 \cdot \left(a \cdot a\right)\right)} \]

   if -6.4999999999999997e110 < rand < 3.39999999999999991e146

   1. Initial program 99.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 88.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 88.1%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 3.39999999999999991e146 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 38.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 38.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]

   13. Taylor expanded in a around inf

       \[\leadsto \color{blue}{27 \cdot {a}^{4}} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \color{blue}{\left({a}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot {a}^{\color{blue}{3}}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 38.5%

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   15. Simplified 38.5%

       \[\leadsto \color{blue}{27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;-0.3333333333333333 + a \cdot \left(1 + \left(a \cdot a\right) \cdot -9\right)\\ \mathbf{elif}\;rand \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -6.5e+110)
   (* (- 0.1111111111111111 (* a a)) (+ (* a 9.0) -3.0))
   (if (<= rand 8.5e+148)
     (+ a -0.3333333333333333)
     (* 27.0 (* a (* a (* a a)))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0);
	} else if (rand <= 8.5e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-6.5d+110)) then
        tmp = (0.1111111111111111d0 - (a * a)) * ((a * 9.0d0) + (-3.0d0))
    else if (rand <= 8.5d+148) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 27.0d0 * (a * (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0);
	} else if (rand <= 8.5e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -6.5e+110:
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0)
	elif rand <= 8.5e+148:
		tmp = a + -0.3333333333333333
	else:
		tmp = 27.0 * (a * (a * (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -6.5e+110)
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(Float64(a * 9.0) + -3.0));
	elseif (rand <= 8.5e+148)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(27.0 * Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -6.5e+110)
		tmp = (0.1111111111111111 - (a * a)) * ((a * 9.0) + -3.0);
	elseif (rand <= 8.5e+148)
		tmp = a + -0.3333333333333333;
	else
		tmp = 27.0 * (a * (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -6.5e+110], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 8.5e+148], N[(a + -0.3333333333333333), $MachinePrecision], N[(27.0 * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -6.4999999999999997e110

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 1.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 1.0%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(9 \cdot a - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(9 \cdot a + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(9 \cdot a\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot 9\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, 9\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       5. metadata-eval 37.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, 9\right), -3\right)\right) \]

   12. Simplified 37.7%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot 9 + -3\right)} \]

   if -6.4999999999999997e110 < rand < 8.4999999999999996e148

   1. Initial program 99.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 88.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 88.1%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 8.4999999999999996e148 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 38.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 38.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]

   13. Taylor expanded in a around inf

       \[\leadsto \color{blue}{27 \cdot {a}^{4}} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \color{blue}{\left({a}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot {a}^{\color{blue}{3}}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 38.5%

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   15. Simplified 38.5%

       \[\leadsto \color{blue}{27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 8.5 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;0.1111111111111111 \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \mathbf{elif}\;rand \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -6.5e+110)
   (* 0.1111111111111111 (+ -3.0 (* a (+ 9.0 (* a -27.0)))))
   (if (<= rand 1.8e+148)
     (+ a -0.3333333333333333)
     (* 27.0 (* a (* a (* a a)))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = 0.1111111111111111 * (-3.0 + (a * (9.0 + (a * -27.0))));
	} else if (rand <= 1.8e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-6.5d+110)) then
        tmp = 0.1111111111111111d0 * ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0)))))
    else if (rand <= 1.8d+148) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 27.0d0 * (a * (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -6.5e+110) {
		tmp = 0.1111111111111111 * (-3.0 + (a * (9.0 + (a * -27.0))));
	} else if (rand <= 1.8e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -6.5e+110:
		tmp = 0.1111111111111111 * (-3.0 + (a * (9.0 + (a * -27.0))))
	elif rand <= 1.8e+148:
		tmp = a + -0.3333333333333333
	else:
		tmp = 27.0 * (a * (a * (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -6.5e+110)
		tmp = Float64(0.1111111111111111 * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))));
	elseif (rand <= 1.8e+148)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(27.0 * Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -6.5e+110)
		tmp = 0.1111111111111111 * (-3.0 + (a * (9.0 + (a * -27.0))));
	elseif (rand <= 1.8e+148)
		tmp = a + -0.3333333333333333;
	else
		tmp = 27.0 * (a * (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -6.5e+110], N[(0.1111111111111111 * N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.8e+148], N[(a + -0.3333333333333333), $MachinePrecision], N[(27.0 * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -6.4999999999999997e110

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 1.0%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 1.0%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 0.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 0.4%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]

   13. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{9}}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]
   14. Step-by-step derivation

   15. Simplified 33.3%

       \[\leadsto \color{blue}{0.1111111111111111} \cdot \left(a \cdot \left(9 + a \cdot -27\right) + -3\right) \]

   if -6.4999999999999997e110 < rand < 1.80000000000000003e148

   1. Initial program 99.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 88.1%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 88.1%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 1.80000000000000003e148 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 38.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 38.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]

   13. Taylor expanded in a around inf

       \[\leadsto \color{blue}{27 \cdot {a}^{4}} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \color{blue}{\left({a}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot {a}^{\color{blue}{3}}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 38.5%

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   15. Simplified 38.5%

       \[\leadsto \color{blue}{27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -6.5 \cdot 10^{+110}:\\ \;\;\;\;0.1111111111111111 \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \mathbf{elif}\;rand \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 1.35e+151)
   (+ a -0.3333333333333333)
   (* 27.0 (* a (* a (* a a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 1.35e+151) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 1.35d+151) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 27.0d0 * (a * (a * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 1.35e+151) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 27.0 * (a * (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 1.35e+151:
		tmp = a + -0.3333333333333333
	else:
		tmp = 27.0 * (a * (a * (a * a)))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 1.35e+151)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(27.0 * Float64(a * Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 1.35e+151)
		tmp = a + -0.3333333333333333;
	else
		tmp = 27.0 * (a * (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 1.35e+151], N[(a + -0.3333333333333333), $MachinePrecision], N[(27.0 * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 1.3500000000000001e151

   1. Initial program 99.8%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 71.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 71.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 1.3500000000000001e151 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 38.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 38.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]

   13. Taylor expanded in a around inf

       \[\leadsto \color{blue}{27 \cdot {a}^{4}} \]
   14. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \color{blue}{\left({a}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \left(a \cdot {a}^{\color{blue}{3}}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 38.5%

           \[\leadsto \mathsf{*.f64}\left(27, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right) \]

   15. Simplified 38.5%

       \[\leadsto \color{blue}{27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;27 \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 67.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 1.5e+151)
   (+ a -0.3333333333333333)
   (* (- 0.1111111111111111 (* a a)) -3.0)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 1.5e+151) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 1.5d+151) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (0.1111111111111111d0 - (a * a)) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 1.5e+151) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 1.5e+151:
		tmp = a + -0.3333333333333333
	else:
		tmp = (0.1111111111111111 - (a * a)) * -3.0
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 1.5e+151)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 1.5e+151)
		tmp = a + -0.3333333333333333;
	else
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 1.5e+151], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 1.5e151

   1. Initial program 99.8%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 71.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 71.9%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

   if 1.5e151 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto a + \frac{-1}{3} \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

      \[\leadsto \color{blue}{-0.3333333333333333 + a} \]
   8. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a}{\color{blue}{\frac{-1}{3} - a}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right) \cdot \color{blue}{\frac{1}{\frac{-1}{3} - a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 36.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{-3}\right) \]
   11. Step-by-step derivation

   12. Simplified 37.4%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{-3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 1.5 \cdot 10^{+151}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.9% accurate, 39.7× speedup

Math

\[\begin{array}{l} \\ a + -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 (+ a -0.3333333333333333))

C

double code(double a, double rand) {
	return a + -0.3333333333333333;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a + (-0.3333333333333333d0)
end function

Java

public static double code(double a, double rand) {
	return a + -0.3333333333333333;
}

Python

def code(a, rand):
	return a + -0.3333333333333333

Julia

function code(a, rand)
	return Float64(a + -0.3333333333333333)
end

MATLAB

function tmp = code(a, rand)
	tmp = a + -0.3333333333333333;
end

Wolfram

code[a_, rand_] := N[(a + -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto a + \frac{-1}{3} \]

   3. +-commutative N/A

      \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

   4. +-lowering-+.f64 64.1%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 64.1%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Final simplification 64.1%

   \[\leadsto a + -0.3333333333333333 \]
9. Add Preprocessing

Alternative 20: 61.8% accurate, 119.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 a)

C

double code(double a, double rand) {
	return a;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a
end function

Java

public static double code(double a, double rand) {
	return a;
}

Python

def code(a, rand):
	return a

Julia

function code(a, rand)
	return a
end

MATLAB

function tmp = code(a, rand)
	tmp = a;
end

Wolfram

code[a_, rand_] := a

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto a + \frac{-1}{3} \]

   3. +-commutative N/A

      \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

   4. +-lowering-+.f64 64.1%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 64.1%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
9. Step-by-step derivation

10. Simplified 62.5%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Alternative 21: 1.5% accurate, 119.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 -0.3333333333333333)

C

double code(double a, double rand) {
	return -0.3333333333333333;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = -0.3333333333333333d0
end function

Java

public static double code(double a, double rand) {
	return -0.3333333333333333;
}

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

function tmp = code(a, rand)
	tmp = -0.3333333333333333;
end

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\left(a + -0.3333333333333333\right) \cdot 9}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto a + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto a + \frac{-1}{3} \]

   3. +-commutative N/A

      \[\leadsto \frac{-1}{3} + \color{blue}{a} \]

   4. +-lowering-+.f64 64.1%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 64.1%

   \[\leadsto \color{blue}{-0.3333333333333333 + a} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{3}} \]
9. Step-by-step derivation

10. Simplified 1.5%

    \[\leadsto \color{blue}{-0.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))))