Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.9%

Time: 7.0s

Alternatives: 10

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: 99.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a rand)
 :precision binary64
 (fma rand (/ (sqrt (- a 0.3333333333333333)) 3.0) (- a 0.3333333333333333)))

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(rand * N[(N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision] / 3.0), $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. 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. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   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%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}, a - 0.3333333333333333, a - 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. sqr-pow N/A

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

   6. sqr-pow N/A

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

   7. pow1/2 N/A

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

   8. lift-fma.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-rgt-in N/A

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

   12. sub-neg N/A

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

   13. associate-*l/ N/A

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

   14. lower-fma.f64 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. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{3}, \sqrt{a - 0.3333333333333333}, a - 0.3333333333333333\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift--.f64 99.9

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

8. Applied rewrites 99.9%

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

Alternative 2: 91.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{if}\;rand \leq -2.9 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 3.9 \cdot 10^{+53}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (* (sqrt a) 0.3333333333333333) rand)))
   (if (<= rand -2.9e+88)
     t_0
     (if (<= rand 3.9e+53) (- a 0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = (sqrt(a) * 0.3333333333333333) * rand;
	double tmp;
	if (rand <= -2.9e+88) {
		tmp = t_0;
	} else if (rand <= 3.9e+53) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(a) * 0.3333333333333333d0) * rand
    if (rand <= (-2.9d+88)) then
        tmp = t_0
    else if (rand <= 3.9d+53) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = (Math.sqrt(a) * 0.3333333333333333) * rand;
	double tmp;
	if (rand <= -2.9e+88) {
		tmp = t_0;
	} else if (rand <= 3.9e+53) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = (math.sqrt(a) * 0.3333333333333333) * rand
	tmp = 0
	if rand <= -2.9e+88:
		tmp = t_0
	elif rand <= 3.9e+53:
		tmp = a - 0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand)
	tmp = 0.0
	if (rand <= -2.9e+88)
		tmp = t_0;
	elseif (rand <= 3.9e+53)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = (sqrt(a) * 0.3333333333333333) * rand;
	tmp = 0.0;
	if (rand <= -2.9e+88)
		tmp = t_0;
	elseif (rand <= 3.9e+53)
		tmp = a - 0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision]}, If[LessEqual[rand, -2.9e+88], t$95$0, If[LessEqual[rand, 3.9e+53], N[(a - 0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.9e88 or 3.89999999999999976e53 < 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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      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%

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

   5. Taylor expanded in a around inf

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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.6%

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

   12. Applied rewrites 93.7%

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

   if -2.9e88 < rand < 3.89999999999999976e53

   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 97.1

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

   5. Applied rewrites 97.1%

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

Alternative 3: 91.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{if}\;rand \leq -2.9 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 3.9 \cdot 10^{+53}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt a) (* 0.3333333333333333 rand))))
   (if (<= rand -2.9e+88)
     t_0
     (if (<= rand 3.9e+53) (- a 0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt(a) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -2.9e+88) {
		tmp = t_0;
	} else if (rand <= 3.9e+53) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(a) * (0.3333333333333333d0 * rand)
    if (rand <= (-2.9d+88)) then
        tmp = t_0
    else if (rand <= 3.9d+53) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt(a) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -2.9e+88) {
		tmp = t_0;
	} else if (rand <= 3.9e+53) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt(a) * (0.3333333333333333 * rand)
	tmp = 0
	if rand <= -2.9e+88:
		tmp = t_0
	elif rand <= 3.9e+53:
		tmp = a - 0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(a) * Float64(0.3333333333333333 * rand))
	tmp = 0.0
	if (rand <= -2.9e+88)
		tmp = t_0;
	elseif (rand <= 3.9e+53)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt(a) * (0.3333333333333333 * rand);
	tmp = 0.0;
	if (rand <= -2.9e+88)
		tmp = t_0;
	elseif (rand <= 3.9e+53)
		tmp = a - 0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[a], $MachinePrecision] * N[(0.3333333333333333 * rand), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -2.9e+88], t$95$0, If[LessEqual[rand, 3.9e+53], N[(a - 0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.9e88 or 3.89999999999999976e53 < 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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      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%

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

   5. Taylor expanded in a around inf

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

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.6%

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

   12. Applied rewrites 93.5%

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

   if -2.9e88 < rand < 3.89999999999999976e53

   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 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.9 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{a} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{elif}\;rand \leq 3.9 \cdot 10^{+53}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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. 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. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   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%

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

6. Applied rewrites 99.8%

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

Alternative 5: 99.8% accurate, 2.4× speedup

Math

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

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

Wolfram

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

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 6: 98.8% accurate, 2.7× speedup

Math

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

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

Wolfram

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

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 98.8%

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

Alternative 7: 67.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 2.25 \cdot 10^{+152}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 2.25e+152)
   (- a 0.3333333333333333)
   (/ (fma a a -0.1111111111111111) 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 2.25e+152) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = fma(a, a, -0.1111111111111111) / 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 2.25e+152)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(fma(a, a, -0.1111111111111111) / 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 2.25e+152], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[(a * a + -0.1111111111111111), $MachinePrecision] / 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 2.25e152

   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 69.2

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

   5. Applied rewrites 69.2%

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

   if 2.25e152 < 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 rand around 0

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

      1. lower--.f64 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 34.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 35.5%

       \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * 0.3333333333333333 + 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. 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. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   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%

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

5. Taylor expanded in a around inf

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

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 98.1

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

7. Applied rewrites 98.1%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 98.2%

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

Alternative 9: 62.4% 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 62.7

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

5. Applied rewrites 62.7%

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

Alternative 10: 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 62.7

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

5. Applied rewrites 62.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024298 
(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))))