Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 11

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{\left(a - 0.3333333333333333\right) \cdot 9}}, a - 0.3333333333333333, a - 0.3333333333333333\right)} \]
5. Add Preprocessing

Alternative 2: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.25 \cdot 10^{+76} \lor \neg \left(rand \leq 6.8 \cdot 10^{+87}\right):\\ \;\;\;\;\left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -1.25e+76) (not (<= rand 6.8e+87)))
   (* (* (sqrt (- a 0.3333333333333333)) 0.3333333333333333) rand)
   (* (/ (- a 0.3333333333333333) rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -1.25e+76) || !(rand <= 6.8e+87)) {
		tmp = (sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand;
	} else {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if ((rand <= (-1.25d+76)) .or. (.not. (rand <= 6.8d+87))) then
        tmp = (sqrt((a - 0.3333333333333333d0)) * 0.3333333333333333d0) * rand
    else
        tmp = ((a - 0.3333333333333333d0) / rand) * rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -1.25e+76) || !(rand <= 6.8e+87)) {
		tmp = (Math.sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand;
	} else {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -1.25e+76) or not (rand <= 6.8e+87):
		tmp = (math.sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand
	else:
		tmp = ((a - 0.3333333333333333) / rand) * rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -1.25e+76) || !(rand <= 6.8e+87))
		tmp = Float64(Float64(sqrt(Float64(a - 0.3333333333333333)) * 0.3333333333333333) * rand);
	else
		tmp = Float64(Float64(Float64(a - 0.3333333333333333) / rand) * rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -1.25e+76) || ~((rand <= 6.8e+87)))
		tmp = (sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand;
	else
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -1.25e+76], N[Not[LessEqual[rand, 6.8e+87]], $MachinePrecision]], N[(N[(N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / rand), $MachinePrecision] * rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.24999999999999998e76 or 6.8000000000000004e87 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 94.7%

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

   if -1.24999999999999998e76 < rand < 6.8000000000000004e87

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in rand around 0

      \[\leadsto \frac{a - \frac{1}{3}}{rand} \cdot rand \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto \frac{a - 0.3333333333333333}{rand} \cdot rand \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.25 \cdot 10^{+76} \lor \neg \left(rand \leq 6.8 \cdot 10^{+87}\right):\\ \;\;\;\;\left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a - 0.3333333333333333}\\ \mathbf{if}\;rand \leq -1.25 \cdot 10^{+76}:\\ \;\;\;\;\left(t\_0 \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{elif}\;rand \leq 6.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot t\_0\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (sqrt (- a 0.3333333333333333))))
   (if (<= rand -1.25e+76)
     (* (* t_0 0.3333333333333333) rand)
     (if (<= rand 6.8e+87)
       (* (/ (- a 0.3333333333333333) rand) rand)
       (* (* rand t_0) 0.3333333333333333)))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a - 0.3333333333333333));
	double tmp;
	if (rand <= -1.25e+76) {
		tmp = (t_0 * 0.3333333333333333) * rand;
	} else if (rand <= 6.8e+87) {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	} else {
		tmp = (rand * t_0) * 0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a - 0.3333333333333333d0))
    if (rand <= (-1.25d+76)) then
        tmp = (t_0 * 0.3333333333333333d0) * rand
    else if (rand <= 6.8d+87) then
        tmp = ((a - 0.3333333333333333d0) / rand) * rand
    else
        tmp = (rand * t_0) * 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a - 0.3333333333333333));
	double tmp;
	if (rand <= -1.25e+76) {
		tmp = (t_0 * 0.3333333333333333) * rand;
	} else if (rand <= 6.8e+87) {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	} else {
		tmp = (rand * t_0) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a - 0.3333333333333333))
	tmp = 0
	if rand <= -1.25e+76:
		tmp = (t_0 * 0.3333333333333333) * rand
	elif rand <= 6.8e+87:
		tmp = ((a - 0.3333333333333333) / rand) * rand
	else:
		tmp = (rand * t_0) * 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	t_0 = sqrt(Float64(a - 0.3333333333333333))
	tmp = 0.0
	if (rand <= -1.25e+76)
		tmp = Float64(Float64(t_0 * 0.3333333333333333) * rand);
	elseif (rand <= 6.8e+87)
		tmp = Float64(Float64(Float64(a - 0.3333333333333333) / rand) * rand);
	else
		tmp = Float64(Float64(rand * t_0) * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a - 0.3333333333333333));
	tmp = 0.0;
	if (rand <= -1.25e+76)
		tmp = (t_0 * 0.3333333333333333) * rand;
	elseif (rand <= 6.8e+87)
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	else
		tmp = (rand * t_0) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[rand, -1.25e+76], N[(N[(t$95$0 * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], If[LessEqual[rand, 6.8e+87], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / rand), $MachinePrecision] * rand), $MachinePrecision], N[(N[(rand * t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.24999999999999998e76

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 93.2%

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

   if -1.24999999999999998e76 < rand < 6.8000000000000004e87

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in rand around 0

      \[\leadsto \frac{a - \frac{1}{3}}{rand} \cdot rand \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto \frac{a - 0.3333333333333333}{rand} \cdot rand \]

   if 6.8000000000000004e87 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 96.5

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

   7. Applied rewrites 96.5%

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

   9. Applied rewrites 96.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.25 \cdot 10^{+76}:\\ \;\;\;\;\left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{elif}\;rand \leq 6.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot \sqrt{a - 0.3333333333333333}\right) \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -2.2 \cdot 10^{+76} \lor \neg \left(rand \leq 6.8 \cdot 10^{+87}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -2.2e+76) (not (<= rand 6.8e+87)))
   (* (* (sqrt a) 0.3333333333333333) rand)
   (* (/ (- a 0.3333333333333333) rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -2.2e+76) || !(rand <= 6.8e+87)) {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if ((rand <= (-2.2d+76)) .or. (.not. (rand <= 6.8d+87))) then
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    else
        tmp = ((a - 0.3333333333333333d0) / rand) * rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -2.2e+76) || !(rand <= 6.8e+87)) {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -2.2e+76) or not (rand <= 6.8e+87):
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	else:
		tmp = ((a - 0.3333333333333333) / rand) * rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -2.2e+76) || !(rand <= 6.8e+87))
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	else
		tmp = Float64(Float64(Float64(a - 0.3333333333333333) / rand) * rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -2.2e+76) || ~((rand <= 6.8e+87)))
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	else
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -2.2e+76], N[Not[LessEqual[rand, 6.8e+87]], $MachinePrecision]], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / rand), $MachinePrecision] * rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.2e76 or 6.8000000000000004e87 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 94.6

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 91.2%

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

   12. Applied rewrites 91.3%

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

   if -2.2e76 < rand < 6.8000000000000004e87

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in rand around 0

      \[\leadsto \frac{a - \frac{1}{3}}{rand} \cdot rand \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto \frac{a - 0.3333333333333333}{rand} \cdot rand \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.2 \cdot 10^{+76} \lor \neg \left(rand \leq 6.8 \cdot 10^{+87}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{elif}\;rand \leq 6.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -2.2e+76)
   (* (* (sqrt a) 0.3333333333333333) rand)
   (if (<= rand 6.8e+87)
     (* (/ (- a 0.3333333333333333) rand) rand)
     (* (* (sqrt a) rand) 0.3333333333333333))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -2.2e+76) {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	} else if (rand <= 6.8e+87) {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	} else {
		tmp = (sqrt(a) * rand) * 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 <= (-2.2d+76)) then
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    else if (rand <= 6.8d+87) then
        tmp = ((a - 0.3333333333333333d0) / rand) * rand
    else
        tmp = (sqrt(a) * rand) * 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -2.2e+76) {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	} else if (rand <= 6.8e+87) {
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	} else {
		tmp = (Math.sqrt(a) * rand) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -2.2e+76:
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	elif rand <= 6.8e+87:
		tmp = ((a - 0.3333333333333333) / rand) * rand
	else:
		tmp = (math.sqrt(a) * rand) * 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -2.2e+76)
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	elseif (rand <= 6.8e+87)
		tmp = Float64(Float64(Float64(a - 0.3333333333333333) / rand) * rand);
	else
		tmp = Float64(Float64(sqrt(a) * rand) * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -2.2e+76)
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	elseif (rand <= 6.8e+87)
		tmp = ((a - 0.3333333333333333) / rand) * rand;
	else
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -2.2e+76], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], If[LessEqual[rand, 6.8e+87], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / rand), $MachinePrecision] * rand), $MachinePrecision], N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.2e76

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 93.2

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 90.4%

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

   12. Applied rewrites 90.5%

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

   if -2.2e76 < rand < 6.8000000000000004e87

   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 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in rand around 0

      \[\leadsto \frac{a - \frac{1}{3}}{rand} \cdot rand \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto \frac{a - 0.3333333333333333}{rand} \cdot rand \]

   if 6.8000000000000004e87 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-fma.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 92.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{elif}\;rand \leq 6.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{rand} \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\sqrt{a - 0.3333333333333333} \cdot rand, \color{blue}{0.3333333333333333}, a - 0.3333333333333333\right) \]
8. Add Preprocessing

Alternative 7: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 99.8

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

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333 \cdot rand, \sqrt{a - 0.3333333333333333}, a - 0.3333333333333333\right)} \]
6. Add Preprocessing

Alternative 8: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 98.1%

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

Alternative 9: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in a around inf

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

10. Applied rewrites 98.1%

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

12. Applied rewrites 98.1%

    \[\leadsto a - \color{blue}{\mathsf{fma}\left(-0.3333333333333333 \cdot rand, \sqrt{a}, 0.3333333333333333\right)} \]
13. Add Preprocessing

Alternative 10: 39.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{a - 0.3333333333333333}{rand} \cdot rand \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return ((a - 0.3333333333333333) / rand) * rand

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. lower-*.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. div-sub N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 69.9

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

5. Applied rewrites 69.9%

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

6. Taylor expanded in rand around 0

   \[\leadsto \frac{a - \frac{1}{3}}{rand} \cdot rand \]
7. Step-by-step derivation

8. Applied rewrites 34.6%

   \[\leadsto \frac{a - 0.3333333333333333}{rand} \cdot rand \]
9. Add Preprocessing

Alternative 11: 38.7% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (a rand) :precision binary64 (* (/ a rand) rand))

C

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

Fortran

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

Java

public static double code(double a, double rand) {
	return (a / rand) * rand;
}

Python

def code(a, rand):
	return (a / rand) * rand

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. lower-*.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. div-sub N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 69.9

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

5. Applied rewrites 69.9%

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

6. Taylor expanded in rand around 0

   \[\leadsto \frac{a - \frac{1}{3}}{rand} \cdot rand \]
7. Step-by-step derivation

8. Applied rewrites 34.6%

   \[\leadsto \frac{a - 0.3333333333333333}{rand} \cdot rand \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{a}{rand} \cdot rand \]
10. Step-by-step derivation

11. Applied rewrites 33.8%

    \[\leadsto \frac{a}{rand} \cdot rand \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))))