Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 12.9s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

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

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

   2. pow1/2 N/A

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

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

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

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

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

   5. +-lowering-+.f64 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -5 \cdot 10^{+64}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 1.16 \cdot 10^{+71}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -5e+64)
   (* rand (* (sqrt (+ a -0.3333333333333333)) 0.3333333333333333))
   (if (<= rand 1.16e+71)
     (+ a -0.3333333333333333)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -5e+64) {
		tmp = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	} else if (rand <= 1.16e+71) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-5d+64)) then
        tmp = rand * (sqrt((a + (-0.3333333333333333d0))) * 0.3333333333333333d0)
    else if (rand <= 1.16d+71) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -5e+64) {
		tmp = rand * (Math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	} else if (rand <= 1.16e+71) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -5e+64:
		tmp = rand * (math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333)
	elif rand <= 1.16e+71:
		tmp = a + -0.3333333333333333
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -5e+64)
		tmp = Float64(rand * Float64(sqrt(Float64(a + -0.3333333333333333)) * 0.3333333333333333));
	elseif (rand <= 1.16e+71)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -5e+64)
		tmp = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	elseif (rand <= 1.16e+71)
		tmp = a + -0.3333333333333333;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -5e+64], N[(rand * N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.16e+71], N[(a + -0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -5e64

   1. Initial program 97.5%

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

   3. Taylor expanded in rand around inf

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

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

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-lowering-+.f64 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in rand around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 82.3%

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

   8. Simplified 82.3%

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

   if -5e64 < rand < 1.1599999999999999e71

   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 98.4%

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

   5. Simplified 98.4%

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

   if 1.1599999999999999e71 < rand

   1. Initial program 99.4%

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

   3. Taylor expanded in 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 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0

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

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

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

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

      4. sqrt-lowering-sqrt.f64 92.8%

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

   8. Simplified 92.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -5 \cdot 10^{+64}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 1.16 \cdot 10^{+71}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.95 \cdot 10^{+65}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.3 \cdot 10^{+69}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.95e+65)
   (* rand (* 0.3333333333333333 (sqrt a)))
   (if (<= rand 1.3e+69)
     (+ a -0.3333333333333333)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.95e+65) {
		tmp = rand * (0.3333333333333333 * sqrt(a));
	} else if (rand <= 1.3e+69) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.95d+65)) then
        tmp = rand * (0.3333333333333333d0 * sqrt(a))
    else if (rand <= 1.3d+69) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.95e+65) {
		tmp = rand * (0.3333333333333333 * Math.sqrt(a));
	} else if (rand <= 1.3e+69) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.95e+65:
		tmp = rand * (0.3333333333333333 * math.sqrt(a))
	elif rand <= 1.3e+69:
		tmp = a + -0.3333333333333333
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.95e+65)
		tmp = Float64(rand * Float64(0.3333333333333333 * sqrt(a)));
	elseif (rand <= 1.3e+69)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.95e+65)
		tmp = rand * (0.3333333333333333 * sqrt(a));
	elseif (rand <= 1.3e+69)
		tmp = a + -0.3333333333333333;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.95e+65], N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.3e+69], N[(a + -0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.9499999999999999e65

   1. Initial program 97.5%

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

   3. Taylor expanded in rand around inf

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

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

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

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

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-lowering-+.f64 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in rand around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 82.3%

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

   8. Simplified 82.3%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 79.9%

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

   11. Simplified 79.9%

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

   if -1.9499999999999999e65 < rand < 1.3000000000000001e69

   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 98.4%

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

   5. Simplified 98.4%

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

   if 1.3000000000000001e69 < rand

   1. Initial program 99.4%

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

   3. Taylor expanded in 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 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0

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

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

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

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

      4. sqrt-lowering-sqrt.f64 92.8%

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

   8. Simplified 92.8%

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

Alternative 4: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.45 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 8 \cdot 10^{+71}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -1.45e+64)
     t_0
     (if (<= rand 8e+71) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -1.45e+64) {
		tmp = t_0;
	} else if (rand <= 8e+71) {
		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 <= (-1.45d+64)) then
        tmp = t_0
    else if (rand <= 8d+71) 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 <= -1.45e+64) {
		tmp = t_0;
	} else if (rand <= 8e+71) {
		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 <= -1.45e+64:
		tmp = t_0
	elif rand <= 8e+71:
		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 <= -1.45e+64)
		tmp = t_0;
	elseif (rand <= 8e+71)
		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 <= -1.45e+64)
		tmp = t_0;
	elseif (rand <= 8e+71)
		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, -1.45e+64], t$95$0, If[LessEqual[rand, 8e+71], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.44999999999999997e64 or 8.0000000000000003e71 < rand

   1. Initial program 98.4%

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

   3. Taylor expanded in 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 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in a around 0

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

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

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

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

      4. sqrt-lowering-sqrt.f64 86.0%

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

   8. Simplified 86.0%

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

   if -1.44999999999999997e64 < rand < 8.0000000000000003e71

   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 98.4%

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

   5. Simplified 98.4%

      \[\leadsto \color{blue}{a + -0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. 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 99.4%

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

3. Taylor expanded in rand around 0

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

   1. +-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: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in 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. *-commutative N/A

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

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

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

   5. sqrt-lowering-sqrt.f64 98.1%

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

8. Simplified 98.1%

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

Alternative 7: 67.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -5.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\frac{rand}{a \cdot a - 0.1111111111111111}} \cdot \frac{rand}{a + 0.3333333333333333}\\ \mathbf{elif}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{rand \cdot \left(a \cdot a + -0.1111111111111111\right)}{rand}}{a + 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -5.6e+156)
   (*
    (/ 1.0 (/ rand (- (* a a) 0.1111111111111111)))
    (/ rand (+ a 0.3333333333333333)))
   (if (<= rand 4.1e+154)
     (+ a -0.3333333333333333)
     (/
      (/ (* rand (+ (* a a) -0.1111111111111111)) rand)
      (+ a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -5.6e+156) {
		tmp = (1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand / (a + 0.3333333333333333));
	} else if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-5.6d+156)) then
        tmp = (1.0d0 / (rand / ((a * a) - 0.1111111111111111d0))) * (rand / (a + 0.3333333333333333d0))
    else if (rand <= 4.1d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = ((rand * ((a * a) + (-0.1111111111111111d0))) / rand) / (a + 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -5.6e+156) {
		tmp = (1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand / (a + 0.3333333333333333));
	} else if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -5.6e+156:
		tmp = (1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand / (a + 0.3333333333333333))
	elif rand <= 4.1e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -5.6e+156)
		tmp = Float64(Float64(1.0 / Float64(rand / Float64(Float64(a * a) - 0.1111111111111111))) * Float64(rand / Float64(a + 0.3333333333333333)));
	elseif (rand <= 4.1e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(rand * Float64(Float64(a * a) + -0.1111111111111111)) / rand) / Float64(a + 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -5.6e+156)
		tmp = (1.0 / (rand / ((a * a) - 0.1111111111111111))) * (rand / (a + 0.3333333333333333));
	elseif (rand <= 4.1e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -5.6e+156], N[(N[(1.0 / N[(rand / N[(N[(a * a), $MachinePrecision] - 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(rand / N[(a + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.1e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(N[(rand * N[(N[(a * a), $MachinePrecision] + -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / rand), $MachinePrecision] / N[(a + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -5.59999999999999975e156

   1. Initial program 94.8%

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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. flip-+ N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

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

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

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

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

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

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

      16. sub-neg N/A

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

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

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

      22. sub-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied egg-rr 0.4%

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

      1. remove-double-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\frac{rand}{a \cdot a + \frac{-1}{9}}\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{rand}{a \cdot a + \frac{-1}{9}}\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\frac{rand}{\mathsf{neg}\left(\left(a \cdot a + \frac{-1}{9}\right)\right)}\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(rand, \left(\mathsf{neg}\left(\left(a \cdot a + \frac{-1}{9}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(rand, \left(\mathsf{neg}\left(\left(\frac{-1}{9} + a \cdot a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(rand, \left(\left(\mathsf{neg}\left(\frac{-1}{9}\right)\right) + \left(\mathsf{neg}\left(a \cdot a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(rand, \left(\frac{1}{9} + \left(\mathsf{neg}\left(a \cdot a\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(rand, \left(\frac{1}{9} - a \cdot a\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(rand, \mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

      11. *-lowering-*.f64 25.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(rand, \mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right)\right)\right)\right), \mathsf{/.f64}\left(rand, \mathsf{+.f64}\left(a, \frac{1}{3}\right)\right)\right) \]

   9. Applied egg-rr 25.7%

      \[\leadsto \frac{1}{\color{blue}{0 - \frac{rand}{0.1111111111111111 - a \cdot a}}} \cdot \frac{rand}{a + 0.3333333333333333} \]

   if -5.59999999999999975e156 < rand < 4.1e154

   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 81.0%

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

   5. Simplified 81.0%

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

   if 4.1e154 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   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 5.6%

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

   5. Simplified 5.6%

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. flip-+ N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

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

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

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

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

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

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

      16. sub-neg N/A

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

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

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

      22. sub-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied egg-rr 34.8%

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-/r* N/A

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

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

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

      9. un-div-inv N/A

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

      10. clear-num N/A

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

      11. /-rgt-identity N/A

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

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

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

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

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

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

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

      15. +-lowering-+.f64 49.6%

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

   9. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{\frac{rand \cdot \left(a \cdot a + -0.1111111111111111\right)}{rand}}{a + 0.3333333333333333}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -5.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\frac{rand}{a \cdot a - 0.1111111111111111}} \cdot \frac{rand}{a + 0.3333333333333333}\\ \mathbf{elif}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{rand \cdot \left(a \cdot a + -0.1111111111111111\right)}{rand}}{a + 0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 3.7 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{rand \cdot \left(a \cdot a + -0.1111111111111111\right)}{rand}}{a + 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 3.7e+154)
   (+ a -0.3333333333333333)
   (/
    (/ (* rand (+ (* a a) -0.1111111111111111)) rand)
    (+ a 0.3333333333333333))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 3.7e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 3.7d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = ((rand * ((a * a) + (-0.1111111111111111d0))) / rand) / (a + 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 3.7e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 3.7e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 3.7e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(rand * Float64(Float64(a * a) + -0.1111111111111111)) / rand) / Float64(a + 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 3.7e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = ((rand * ((a * a) + -0.1111111111111111)) / rand) / (a + 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 3.7e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(N[(rand * N[(N[(a * a), $MachinePrecision] + -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / rand), $MachinePrecision] / N[(a + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 3.69999999999999994e154

   1. Initial program 99.4%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   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 73.5%

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

   5. Simplified 73.5%

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

   if 3.69999999999999994e154 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   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 5.6%

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

   5. Simplified 5.6%

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. flip-+ N/A

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

      11. associate-/r/ N/A

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

      12. times-frac N/A

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

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

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

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

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

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

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

      16. sub-neg N/A

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

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

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

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

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

      22. sub-neg N/A

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

      23. metadata-eval N/A

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

   7. Applied egg-rr 34.8%

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-/r* N/A

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

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

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

      9. un-div-inv N/A

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

      10. clear-num N/A

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

      11. /-rgt-identity N/A

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

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

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

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

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

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

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

      15. +-lowering-+.f64 49.6%

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

   9. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{\frac{rand \cdot \left(a \cdot a + -0.1111111111111111\right)}{rand}}{a + 0.3333333333333333}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot rand\right) \cdot \frac{1}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 3.3e+154) (+ a -0.3333333333333333) (* (* a rand) (/ 1.0 rand))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 3.3e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) * (1.0 / rand);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 3.3d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (a * rand) * (1.0d0 / rand)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 3.3e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) * (1.0 / rand);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 3.3e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = (a * rand) * (1.0 / rand)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 3.3e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(a * rand) * Float64(1.0 / rand));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 3.3e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = (a * rand) * (1.0 / rand);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 3.3e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(a * rand), $MachinePrecision] * N[(1.0 / rand), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 3.3e154

   1. Initial program 99.4%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   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 73.5%

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

   5. Simplified 73.5%

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

   if 3.3e154 < rand

   1. Initial program 99.5%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. 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 5.6%

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

   5. Simplified 5.6%

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. associate-*r* N/A

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

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

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

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

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

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

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

      14. /-lowering-/.f64 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 49.2%

       \[\leadsto \left(rand \cdot \color{blue}{a}\right) \cdot \frac{1}{rand} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot rand\right) \cdot \frac{1}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 8 \cdot 10^{+145}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 8e+145) (+ a -0.3333333333333333) (* 9.0 (* a (* a a)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 8e+145) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 9.0 * (a * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 8d+145) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 9.0d0 * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 8e+145) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 9.0 * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 8e+145:
		tmp = a + -0.3333333333333333
	else:
		tmp = 9.0 * (a * (a * a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 8e+145)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(9.0 * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 8e+145)
		tmp = a + -0.3333333333333333;
	else
		tmp = 9.0 * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 8e+145], N[(a + -0.3333333333333333), $MachinePrecision], N[(9.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 7.9999999999999999e145

   1. Initial program 99.4%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   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 74.7%

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

   5. Simplified 74.7%

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

   if 7.9999999999999999e145 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   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. flip3-+ N/A

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

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

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

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

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

      4. cube-mult N/A

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

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

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

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

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. *-lowering-*.f64 14.1%

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

   7. Applied egg-rr 14.1%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 43.9%

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

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{9 \cdot {a}^{3}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(9, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(9, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(9, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

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

       6. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(9, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   13. Simplified 43.9%

       \[\leadsto \color{blue}{9 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.0% 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 99.4%

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

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
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 65.8%

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

5. Simplified 65.8%

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

Alternative 12: 61.0% 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 99.4%

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

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
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 65.8%

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

5. Simplified 65.8%

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

6. Taylor expanded in a around inf

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

8. Simplified 64.6%

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

Alternative 13: 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 99.4%

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

3. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
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 65.8%

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

5. Simplified 65.8%

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

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 2024191 
(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))))