Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 12.0s

Alternatives: 18

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{\frac{rand}{{\left(a + -0.3333333333333333\right)}^{0.5}}}{3}\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (*
  (+ a -0.3333333333333333)
  (+ 1.0 (/ (/ rand (pow (+ a -0.3333333333333333) 0.5)) 3.0))))

C

double code(double a, double rand) {
	return (a + -0.3333333333333333) * (1.0 + ((rand / pow((a + -0.3333333333333333), 0.5)) / 3.0));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a + (-0.3333333333333333d0)) * (1.0d0 + ((rand / ((a + (-0.3333333333333333d0)) ** 0.5d0)) / 3.0d0))
end function

Java

public static double code(double a, double rand) {
	return (a + -0.3333333333333333) * (1.0 + ((rand / Math.pow((a + -0.3333333333333333), 0.5)) / 3.0));
}

Python

def code(a, rand):
	return (a + -0.3333333333333333) * (1.0 + ((rand / math.pow((a + -0.3333333333333333), 0.5)) / 3.0))

Julia

function code(a, rand)
	return Float64(Float64(a + -0.3333333333333333) * Float64(1.0 + Float64(Float64(rand / (Float64(a + -0.3333333333333333) ^ 0.5)) / 3.0)))
end

MATLAB

function tmp = code(a, rand)
	tmp = (a + -0.3333333333333333) * (1.0 + ((rand / ((a + -0.3333333333333333) ^ 0.5)) / 3.0));
end

Wolfram

code[a_, rand_] := N[(N[(a + -0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(N[(rand / N[Power[N[(a + -0.3333333333333333), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. *-lft-identity N/A

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

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

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

4. Applied egg-rr 99.9%

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

Alternative 2: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* (sqrt (+ a -0.3333333333333333)) 0.3333333333333333))))
   (if (<= rand -7.5e+66)
     t_0
     (if (<= rand 4.5e+87) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	double tmp;
	if (rand <= -7.5e+66) {
		tmp = t_0;
	} else if (rand <= 4.5e+87) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = rand * (sqrt((a + (-0.3333333333333333d0))) * 0.3333333333333333d0)
    if (rand <= (-7.5d+66)) then
        tmp = t_0
    else if (rand <= 4.5d+87) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand * (Math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	double tmp;
	if (rand <= -7.5e+66) {
		tmp = t_0;
	} else if (rand <= 4.5e+87) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333)
	tmp = 0
	if rand <= -7.5e+66:
		tmp = t_0
	elif rand <= 4.5e+87:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(sqrt(Float64(a + -0.3333333333333333)) * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -7.5e+66)
		tmp = t_0;
	elseif (rand <= 4.5e+87)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -7.5e+66)
		tmp = t_0;
	elseif (rand <= 4.5e+87)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -7.5e+66], t$95$0, If[LessEqual[rand, 4.5e+87], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -7.50000000000000024e66 or 4.5000000000000003e87 < rand

   1. Initial program 98.7%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 90.5%

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

   5. Simplified 90.5%

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 90.6%

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

   7. Applied egg-rr 90.6%

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

   if -7.50000000000000024e66 < rand < 4.5000000000000003e87

   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. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 95.2%

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

   5. Simplified 95.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2 \cdot 10^{+87}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ a -0.3333333333333333)) (* rand 0.3333333333333333))))
   (if (<= rand -7.5e+66)
     t_0
     (if (<= rand 2e+87) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -7.5e+66) {
		tmp = t_0;
	} else if (rand <= 2e+87) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a + (-0.3333333333333333d0))) * (rand * 0.3333333333333333d0)
    if (rand <= (-7.5d+66)) then
        tmp = t_0
    else if (rand <= 2d+87) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -7.5e+66) {
		tmp = t_0;
	} else if (rand <= 2e+87) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333)
	tmp = 0
	if rand <= -7.5e+66:
		tmp = t_0
	elif rand <= 2e+87:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(a + -0.3333333333333333)) * Float64(rand * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -7.5e+66)
		tmp = t_0;
	elseif (rand <= 2e+87)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a + -0.3333333333333333)) * (rand * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -7.5e+66)
		tmp = t_0;
	elseif (rand <= 2e+87)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -7.5e+66], t$95$0, If[LessEqual[rand, 2e+87], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -7.50000000000000024e66 or 1.9999999999999999e87 < rand

   1. Initial program 98.7%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 90.5%

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

   5. Simplified 90.5%

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

   if -7.50000000000000024e66 < rand < 1.9999999999999999e87

   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. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 95.2%

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

   5. Simplified 95.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 2 \cdot 10^{+87}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -6.4 \cdot 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+86}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -6.4e+66)
     t_0
     (if (<= rand 9.6e+86) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -6.4e+66) {
		tmp = t_0;
	} else if (rand <= 9.6e+86) {
		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 <= (-6.4d+66)) then
        tmp = t_0
    else if (rand <= 9.6d+86) 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 <= -6.4e+66) {
		tmp = t_0;
	} else if (rand <= 9.6e+86) {
		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 <= -6.4e+66:
		tmp = t_0
	elif rand <= 9.6e+86:
		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 <= -6.4e+66)
		tmp = t_0;
	elseif (rand <= 9.6e+86)
		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 <= -6.4e+66)
		tmp = t_0;
	elseif (rand <= 9.6e+86)
		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, -6.4e+66], t$95$0, If[LessEqual[rand, 9.6e+86], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -6.3999999999999999e66 or 9.6000000000000001e86 < rand

   1. Initial program 98.7%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)} \]
   7. 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. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 88.8%

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

   8. Simplified 88.8%

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

   if -6.3999999999999999e66 < rand < 9.6000000000000001e86

   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. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 95.2%

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

   5. Simplified 95.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -6.4 \cdot 10^{+66}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 9.6 \cdot 10^{+86}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in rand around 0

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

   1. +-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. +-commutative N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

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

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 6: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in rand around 0

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

   1. +-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. +-commutative N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

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

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

   15. *-lowering-*.f64 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in a around inf

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 7: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)} \]
4. 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 98.1%

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

5. Simplified 98.1%

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

   1. *-commutative N/A

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

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

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

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

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

   4. *-commutative N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. div-inv N/A

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

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

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

   13. sqrt-lowering-sqrt.f64 98.1%

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

7. Applied egg-rr 98.1%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 98.1%

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

9. Applied egg-rr 98.1%

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

10. Final simplification 98.1%

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

Alternative 8: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)} \]
4. 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 98.1%

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

5. Simplified 98.1%

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

   1. *-commutative N/A

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

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

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

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

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

   4. *-commutative N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. div-inv N/A

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

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

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

   13. sqrt-lowering-sqrt.f64 98.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 9: 97.8% 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 99.5%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)} \]
4. 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 98.1%

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

5. Simplified 98.1%

   \[\leadsto \color{blue}{a \cdot \left(1 + \sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right)\right)} \]
6. 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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. pow1/2 N/A

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

   8. inv-pow N/A

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

   9. pow-pow N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/2 N/A

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

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

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

   15. *-commutative N/A

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

   16. metadata-eval N/A

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

   17. div-inv N/A

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

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

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

   19. sqrt-lowering-sqrt.f64 98.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 10: 97.7% 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 99.5%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)} \]
4. 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 98.1%

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

5. Simplified 98.1%

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

6. Taylor expanded in a around 0

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

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

   4. sqrt-lowering-sqrt.f64 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 11: 74.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.1111111111111111 - a \cdot a\\ \mathbf{if}\;rand \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3\right)\\ \mathbf{elif}\;rand \leq 3.4 \cdot 10^{+152}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- 0.1111111111111111 (* a a))))
   (if (<= rand -1.75e+130)
     (* t_0 (+ (* a (+ 9.0 (* a (+ -27.0 (* a 81.0))))) -3.0))
     (if (<= rand 3.4e+152)
       (+ a -0.3333333333333333)
       (* t_0 (+ -3.0 (* a (+ 9.0 (* a -27.0)))))))))

C

double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -1.75e+130) {
		tmp = t_0 * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	} else if (rand <= 3.4e+152) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.1111111111111111d0 - (a * a)
    if (rand <= (-1.75d+130)) then
        tmp = t_0 * ((a * (9.0d0 + (a * ((-27.0d0) + (a * 81.0d0))))) + (-3.0d0))
    else if (rand <= 3.4d+152) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -1.75e+130) {
		tmp = t_0 * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	} else if (rand <= 3.4e+152) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.1111111111111111 - (a * a)
	tmp = 0
	if rand <= -1.75e+130:
		tmp = t_0 * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0)
	elif rand <= 3.4e+152:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.1111111111111111 - Float64(a * a))
	tmp = 0.0
	if (rand <= -1.75e+130)
		tmp = Float64(t_0 * Float64(Float64(a * Float64(9.0 + Float64(a * Float64(-27.0 + Float64(a * 81.0))))) + -3.0));
	elseif (rand <= 3.4e+152)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.1111111111111111 - (a * a);
	tmp = 0.0;
	if (rand <= -1.75e+130)
		tmp = t_0 * ((a * (9.0 + (a * (-27.0 + (a * 81.0))))) + -3.0);
	elseif (rand <= 3.4e+152)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.75e+130], N[(t$95$0 * 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, 3.4e+152], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.75e130

   1. Initial program 99.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 0.4%

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

   5. Simplified 0.4%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 0.4%

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

   7. Applied egg-rr 0.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + a \cdot \left(81 \cdot a - 27\right)\right) - 3\right)}\right) \]
   9. 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 43.3%

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

   10. Simplified 43.3%

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

   if -1.75e130 < rand < 3.4000000000000002e152

   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. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 85.7%

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

   5. Simplified 85.7%

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

   if 3.4000000000000002e152 < rand

   1. Initial program 97.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 5.7%

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

   5. Simplified 5.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 32.2%

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

   7. Applied egg-rr 32.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   9. 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 41.0%

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

   10. Simplified 41.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot \left(9 + a \cdot \left(-27 + a \cdot 81\right)\right) + -3\right)\\ \mathbf{elif}\;rand \leq 3.4 \cdot 10^{+152}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.1111111111111111 - a \cdot a\\ \mathbf{if}\;rand \leq -2.7 \cdot 10^{+130}:\\ \;\;\;\;t\_0 \cdot \left(-3 + a \cdot 9\right)\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+149}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- 0.1111111111111111 (* a a))))
   (if (<= rand -2.7e+130)
     (* t_0 (+ -3.0 (* a 9.0)))
     (if (<= rand 1.35e+149)
       (+ a -0.3333333333333333)
       (* t_0 (+ -3.0 (* a (+ 9.0 (* a -27.0)))))))))

C

double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -2.7e+130) {
		tmp = t_0 * (-3.0 + (a * 9.0));
	} else if (rand <= 1.35e+149) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.1111111111111111d0 - (a * a)
    if (rand <= (-2.7d+130)) then
        tmp = t_0 * ((-3.0d0) + (a * 9.0d0))
    else if (rand <= 1.35d+149) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -2.7e+130) {
		tmp = t_0 * (-3.0 + (a * 9.0));
	} else if (rand <= 1.35e+149) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.1111111111111111 - (a * a)
	tmp = 0
	if rand <= -2.7e+130:
		tmp = t_0 * (-3.0 + (a * 9.0))
	elif rand <= 1.35e+149:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.1111111111111111 - Float64(a * a))
	tmp = 0.0
	if (rand <= -2.7e+130)
		tmp = Float64(t_0 * Float64(-3.0 + Float64(a * 9.0)));
	elseif (rand <= 1.35e+149)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.1111111111111111 - (a * a);
	tmp = 0.0;
	if (rand <= -2.7e+130)
		tmp = t_0 * (-3.0 + (a * 9.0));
	elseif (rand <= 1.35e+149)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -2.7e+130], N[(t$95$0 * N[(-3.0 + N[(a * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.35e+149], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.6999999999999998e130

   1. Initial program 99.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 0.4%

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

   5. Simplified 0.4%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 0.4%

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

   7. Applied egg-rr 0.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(9 \cdot a - 3\right)}\right) \]
   9. 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 43.2%

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

   10. Simplified 43.2%

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

   if -2.6999999999999998e130 < rand < 1.35e149

   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. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 85.7%

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

   5. Simplified 85.7%

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

   if 1.35e149 < rand

   1. Initial program 97.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 5.7%

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

   5. Simplified 5.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 32.2%

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

   7. Applied egg-rr 32.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   9. 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 41.0%

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

   10. Simplified 41.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.7 \cdot 10^{+130}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot 9\right)\\ \mathbf{elif}\;rand \leq 1.35 \cdot 10^{+149}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -2.15 \cdot 10^{+130}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot 9\right)\\ \mathbf{elif}\;rand \leq 4.7 \cdot 10^{+150}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot a\right) + -0.037037037037037035}{0.1111111111111111}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -2.15e+130)
   (* (- 0.1111111111111111 (* a a)) (+ -3.0 (* a 9.0)))
   (if (<= rand 4.7e+150)
     (+ a -0.3333333333333333)
     (/ (+ (* a (* a a)) -0.037037037037037035) 0.1111111111111111))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -2.15e+130) {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0));
	} else if (rand <= 4.7e+150) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-2.15d+130)) then
        tmp = (0.1111111111111111d0 - (a * a)) * ((-3.0d0) + (a * 9.0d0))
    else if (rand <= 4.7d+150) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = ((a * (a * a)) + (-0.037037037037037035d0)) / 0.1111111111111111d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -2.15e+130) {
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0));
	} else if (rand <= 4.7e+150) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -2.15e+130:
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0))
	elif rand <= 4.7e+150:
		tmp = a + -0.3333333333333333
	else:
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -2.15e+130)
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * Float64(-3.0 + Float64(a * 9.0)));
	elseif (rand <= 4.7e+150)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(a * Float64(a * a)) + -0.037037037037037035) / 0.1111111111111111);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -2.15e+130)
		tmp = (0.1111111111111111 - (a * a)) * (-3.0 + (a * 9.0));
	elseif (rand <= 4.7e+150)
		tmp = a + -0.3333333333333333;
	else
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -2.15e+130], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-3.0 + N[(a * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.7e+150], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -0.037037037037037035), $MachinePrecision] / 0.1111111111111111), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.14999999999999992e130

   1. Initial program 99.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 0.4%

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

   5. Simplified 0.4%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 0.4%

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

   7. Applied egg-rr 0.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(9 \cdot a - 3\right)}\right) \]
   9. 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 43.2%

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

   10. Simplified 43.2%

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

   if -2.14999999999999992e130 < rand < 4.70000000000000004e150

   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. Add Preprocessing

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 85.7%

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

   5. Simplified 85.7%

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

   if 4.70000000000000004e150 < rand

   1. Initial program 97.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 5.7%

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

   5. Simplified 5.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 32.2%

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

   7. Applied egg-rr 32.2%

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

      1. un-div-inv N/A

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

      2. metadata-eval N/A

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

      3. flip-+ N/A

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

      4. +-commutative N/A

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

      5. flip3-+ N/A

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

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

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. associate-+r- N/A

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

      14. +-commutative N/A

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

      15. associate-+r- N/A

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

      16. *-commutative N/A

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

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

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

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

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

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

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

      20. sub-neg N/A

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

      21. metadata-eval N/A

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

      22. +-lowering-+.f64 9.6%

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

   9. Applied egg-rr 9.6%

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

   10. Taylor expanded in a around 0

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

   12. Simplified 40.4%

       \[\leadsto \frac{a \cdot \left(a \cdot a\right) + -0.037037037037037035}{\color{blue}{0.1111111111111111}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.15 \cdot 10^{+130}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot 9\right)\\ \mathbf{elif}\;rand \leq 4.7 \cdot 10^{+150}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot a\right) + -0.037037037037037035}{0.1111111111111111}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 8.5 \cdot 10^{+148}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot a\right) + -0.037037037037037035}{0.1111111111111111}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 8.5e+148)
   (+ a -0.3333333333333333)
   (/ (+ (* a (* a a)) -0.037037037037037035) 0.1111111111111111)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 8.5e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 8.5d+148) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = ((a * (a * a)) + (-0.037037037037037035d0)) / 0.1111111111111111d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 8.5e+148) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 8.5e+148:
		tmp = a + -0.3333333333333333
	else:
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 8.5e+148)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(a * Float64(a * a)) + -0.037037037037037035) / 0.1111111111111111);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 8.5e+148)
		tmp = a + -0.3333333333333333;
	else
		tmp = ((a * (a * a)) + -0.037037037037037035) / 0.1111111111111111;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 8.5e+148], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + -0.037037037037037035), $MachinePrecision] / 0.1111111111111111), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 8.4999999999999996e148

   1. Initial program 99.8%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 73.4%

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

   5. Simplified 73.4%

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

   if 8.4999999999999996e148 < rand

   1. Initial program 97.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 5.7%

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

   5. Simplified 5.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 32.2%

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

   7. Applied egg-rr 32.2%

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

      1. un-div-inv N/A

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

      2. metadata-eval N/A

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

      3. flip-+ N/A

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

      4. +-commutative N/A

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

      5. flip3-+ N/A

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

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

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

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

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

      8. cube-mult N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. associate-+r- N/A

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

      14. +-commutative N/A

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

      15. associate-+r- N/A

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

      16. *-commutative N/A

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

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

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

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

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

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

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

      20. sub-neg N/A

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

      21. metadata-eval N/A

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

      22. +-lowering-+.f64 9.6%

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

   9. Applied egg-rr 9.6%

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

   10. Taylor expanded in a around 0

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

   12. Simplified 40.4%

       \[\leadsto \frac{a \cdot \left(a \cdot a\right) + -0.037037037037037035}{\color{blue}{0.1111111111111111}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 2.7e+141)
   (+ a -0.3333333333333333)
   (* (- 0.1111111111111111 (* a a)) -3.0)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 2.7e+141) {
		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 <= 2.7d+141) 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 <= 2.7e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 2.7e+141:
		tmp = a + -0.3333333333333333
	else:
		tmp = (0.1111111111111111 - (a * a)) * -3.0
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 2.7e+141)
		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 <= 2.7e+141)
		tmp = a + -0.3333333333333333;
	else
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 2.7e+141], 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 < 2.7000000000000001e141

   1. Initial program 99.8%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 73.4%

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

   5. Simplified 73.4%

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

   if 2.7000000000000001e141 < rand

   1. Initial program 97.6%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   4. 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. +-lowering-+.f64 5.7%

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

   5. Simplified 5.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. --lowering--.f64 32.2%

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

   7. Applied egg-rr 32.2%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 33.3%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{-3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.8% 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 99.5%

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

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
4. 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. +-lowering-+.f64 62.9%

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

5. Simplified 62.9%

   \[\leadsto \color{blue}{a + -0.3333333333333333} \]
6. Add Preprocessing

Alternative 17: 61.7% 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 99.5%

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

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
4. 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. +-lowering-+.f64 62.9%

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

5. Simplified 62.9%

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

6. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
7. Step-by-step derivation

8. Simplified 61.9%

   \[\leadsto \color{blue}{a} \]
9. Add Preprocessing

Alternative 18: 1.6% accurate, 119.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 -0.3333333333333333)

C

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

Fortran

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

Java

public static double code(double a, double rand) {
	return -0.3333333333333333;
}

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
4. 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. +-lowering-+.f64 62.9%

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

5. Simplified 62.9%

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

6. Taylor expanded in a around 0

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

8. Simplified 1.5%

   \[\leadsto \color{blue}{-0.3333333333333333} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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))))