Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 7.9s

Alternatives: 7

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: 99.7% 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, 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 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.45 \cdot 10^{+72} \lor \neg \left(rand \leq 4.6 \cdot 10^{+86}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -1.45e+72) (not (<= rand 4.6e+86)))
   (* (* (sqrt a) rand) 0.3333333333333333)
   (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -1.45e+72) || !(rand <= 4.6e+86)) {
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	} else {
		tmp = 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 <= (-1.45d+72)) .or. (.not. (rand <= 4.6d+86))) then
        tmp = (sqrt(a) * rand) * 0.3333333333333333d0
    else
        tmp = a - 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -1.45e+72) || !(rand <= 4.6e+86)) {
		tmp = (Math.sqrt(a) * rand) * 0.3333333333333333;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -1.45e+72) or not (rand <= 4.6e+86):
		tmp = (math.sqrt(a) * rand) * 0.3333333333333333
	else:
		tmp = a - 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -1.45e+72) || !(rand <= 4.6e+86))
		tmp = Float64(Float64(sqrt(a) * rand) * 0.3333333333333333);
	else
		tmp = Float64(a - 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -1.45e+72) || ~((rand <= 4.6e+86)))
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	else
		tmp = a - 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -1.45e+72], N[Not[LessEqual[rand, 4.6e+86]], $MachinePrecision]], N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(a - 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.45000000000000009e72 or 4.59999999999999979e86 < rand

   1. Initial program 99.6%

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

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 92.5%

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

   if -1.45000000000000009e72 < rand < 4.59999999999999979e86

   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. lower--.f64 95.6

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

   5. Applied rewrites 95.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.45 \cdot 10^{+72} \lor \neg \left(rand \leq 4.6 \cdot 10^{+86}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.45 \cdot 10^{+72}:\\ \;\;\;\;\left(0.3333333333333333 \cdot rand\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 4.6 \cdot 10^{+86}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.45e+72)
   (* (* 0.3333333333333333 rand) (sqrt a))
   (if (<= rand 4.6e+86)
     (- a 0.3333333333333333)
     (* (* (sqrt a) rand) 0.3333333333333333))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.45e+72) {
		tmp = (0.3333333333333333 * rand) * sqrt(a);
	} else if (rand <= 4.6e+86) {
		tmp = a - 0.3333333333333333;
	} 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 <= (-1.45d+72)) then
        tmp = (0.3333333333333333d0 * rand) * sqrt(a)
    else if (rand <= 4.6d+86) then
        tmp = a - 0.3333333333333333d0
    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 <= -1.45e+72) {
		tmp = (0.3333333333333333 * rand) * Math.sqrt(a);
	} else if (rand <= 4.6e+86) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = (Math.sqrt(a) * rand) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.45e+72:
		tmp = (0.3333333333333333 * rand) * math.sqrt(a)
	elif rand <= 4.6e+86:
		tmp = a - 0.3333333333333333
	else:
		tmp = (math.sqrt(a) * rand) * 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.45e+72)
		tmp = Float64(Float64(0.3333333333333333 * rand) * sqrt(a));
	elseif (rand <= 4.6e+86)
		tmp = Float64(a - 0.3333333333333333);
	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 <= -1.45e+72)
		tmp = (0.3333333333333333 * rand) * sqrt(a);
	elseif (rand <= 4.6e+86)
		tmp = a - 0.3333333333333333;
	else
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.45e+72], N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.6e+86], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.45000000000000009e72

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      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 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 92.3%

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

   if -1.45000000000000009e72 < rand < 4.59999999999999979e86

   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. lower--.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 4.59999999999999979e86 < rand

   1. Initial program 99.7%

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

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 92.8%

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

Alternative 4: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[a], $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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 98.6%

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

Alternative 5: 97.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 97.8

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 97.9%

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

10. Applied rewrites 97.9%

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

Alternative 6: 62.5% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. lower--.f64 59.7

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

5. Applied rewrites 59.7%

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

Alternative 7: 1.5% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. lower--.f64 59.7

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

5. Applied rewrites 59.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 1.5%

   \[\leadsto -0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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))))