Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.7%

Time: 11.3s

Alternatives: 7

Speedup: 1.6×

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: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(rand, \sqrt{\frac{1}{\mathsf{fma}\left(a, 9, -3\right)}}, 1\right) \cdot \left(a + -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (* (fma rand (sqrt (/ 1.0 (fma a 9.0 -3.0))) 1.0) (+ a -0.3333333333333333)))

C

double code(double a, double rand) {
	return fma(rand, sqrt((1.0 / fma(a, 9.0, -3.0))), 1.0) * (a + -0.3333333333333333);
}

Julia

function code(a, rand)
	return Float64(fma(rand, sqrt(Float64(1.0 / fma(a, 9.0, -3.0))), 1.0) * Float64(a + -0.3333333333333333))
end

Wolfram

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

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

   7. unpow2 N/A

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

   8. rem-square-sqrt 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft1-in N/A

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

   7. rgt-mult-inverse N/A

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

   8. distribute-lft-in N/A

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

   9. lower-*.f64 N/A

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

8. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(rand, \sqrt{\frac{1}{\mathsf{fma}\left(a, 9, -3\right)}}, 1\right) \cdot \left(a + -0.3333333333333333\right)} \]
9. Add Preprocessing

Alternative 2: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(a + -0.3333333333333333\right) \cdot \mathsf{fma}\left(rand, \sqrt{\frac{0.1111111111111111}{a}}, 1\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (* (+ a -0.3333333333333333) (fma rand (sqrt (/ 0.1111111111111111 a)) 1.0)))

C

double code(double a, double rand) {
	return (a + -0.3333333333333333) * fma(rand, sqrt((0.1111111111111111 / a)), 1.0);
}

Julia

function code(a, rand)
	return Float64(Float64(a + -0.3333333333333333) * fma(rand, sqrt(Float64(0.1111111111111111 / a)), 1.0))
end

Wolfram

code[a_, rand_] := N[(N[(a + -0.3333333333333333), $MachinePrecision] * N[(rand * N[Sqrt[N[(0.1111111111111111 / a), $MachinePrecision]], $MachinePrecision] + 1.0), $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 a around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

   7. unpow2 N/A

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

   8. rem-square-sqrt 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft1-in N/A

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

   7. rgt-mult-inverse N/A

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

   8. distribute-lft-in N/A

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

   9. lower-*.f64 N/A

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

8. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(rand, \sqrt{\frac{1}{\mathsf{fma}\left(a, 9, -3\right)}}, 1\right) \cdot \left(a + -0.3333333333333333\right)} \]

9. Taylor expanded in a around inf

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

    1. lower-/.f64 99.1

       \[\leadsto \mathsf{fma}\left(rand, \sqrt{\color{blue}{\frac{0.1111111111111111}{a}}}, 1\right) \cdot \left(a + -0.3333333333333333\right) \]

11. Simplified 99.1%

    \[\leadsto \mathsf{fma}\left(rand, \sqrt{\color{blue}{\frac{0.1111111111111111}{a}}}, 1\right) \cdot \left(a + -0.3333333333333333\right) \]

12. Final simplification 99.1%

    \[\leadsto \left(a + -0.3333333333333333\right) \cdot \mathsf{fma}\left(rand, \sqrt{\frac{0.1111111111111111}{a}}, 1\right) \]
13. Add Preprocessing

Alternative 3: 90.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.3 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -1.3e+45)
     t_0
     (if (<= rand 6.5e+76) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -1.3e+45) {
		tmp = t_0;
	} else if (rand <= 6.5e+76) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (rand * sqrt(a))
    if (rand <= (-1.3d+45)) then
        tmp = t_0
    else if (rand <= 6.5d+76) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * Math.sqrt(a));
	double tmp;
	if (rand <= -1.3e+45) {
		tmp = t_0;
	} else if (rand <= 6.5e+76) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.3333333333333333 * (rand * math.sqrt(a))
	tmp = 0
	if rand <= -1.3e+45:
		tmp = t_0
	elif rand <= 6.5e+76:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.3333333333333333 * Float64(rand * sqrt(a)))
	tmp = 0.0
	if (rand <= -1.3e+45)
		tmp = t_0;
	elseif (rand <= 6.5e+76)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.3333333333333333 * (rand * sqrt(a));
	tmp = 0.0;
	if (rand <= -1.3e+45)
		tmp = t_0;
	elseif (rand <= 6.5e+76)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.3e+45], t$95$0, If[LessEqual[rand, 6.5e+76], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.30000000000000004e45 or 6.5000000000000005e76 < 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 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

         \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

      7. unpow2 N/A

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

      8. rem-square-sqrt 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.6

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

   8. Simplified 98.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 89.9

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

   11. Simplified 89.9%

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

   if -1.30000000000000004e45 < rand < 6.5000000000000005e76

   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 a around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

         \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

      7. unpow2 N/A

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

      8. rem-square-sqrt 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 96.9

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

   8. Simplified 96.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.3 \cdot 10^{+45}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $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 a around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

   7. unpow2 N/A

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

   8. rem-square-sqrt 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft1-in N/A

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

   7. rgt-mult-inverse N/A

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

   8. distribute-lft-in N/A

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

   9. lower-*.f64 N/A

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

8. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(rand, \sqrt{\frac{1}{\mathsf{fma}\left(a, 9, -3\right)}}, 1\right) \cdot \left(a + -0.3333333333333333\right)} \]

9. Taylor expanded in rand around 0

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. associate-*l* N/A

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

    4. lower-fma.f64 N/A

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

11. Simplified 99.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(rand, \sqrt{\frac{1}{\mathsf{fma}\left(a, 9, -3\right)}} \cdot \left(-0.3333333333333333 + a\right), -0.3333333333333333 + a\right)} \]

12. Taylor expanded in a around inf

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 99.1

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 5: 97.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $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 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

   7. unpow2 N/A

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

   8. rem-square-sqrt 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.0

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

8. Simplified 98.0%

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

9. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-sqrt.f64 98.0

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

11. Simplified 98.0%

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

12. Final simplification 98.0%

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

Alternative 6: 63.2% 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 a around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

   7. unpow2 N/A

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

   8. rem-square-sqrt 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 62.3

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

8. Simplified 62.3%

   \[\leadsto \color{blue}{a + -0.3333333333333333} \]
9. 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 a around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{a \cdot 9 + \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot rand\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(a, 9, {\left(\sqrt{-3}\right)}^{2}\right)}}} \cdot rand\right) \]

   7. unpow2 N/A

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

   8. rem-square-sqrt 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 62.3

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

8. Simplified 62.3%

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

9. Taylor expanded in a around 0

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

11. Simplified 1.5%

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

Reproduce

herbie shell --seed 2024215 
(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))))