Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.8%

Time: 11.7s

Alternatives: 13

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft1-in N/A

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lift-+.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 91.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.65 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.05 \cdot 10^{+73}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* 0.3333333333333333 (sqrt a)))))
   (if (<= rand -1.65e+47)
     t_0
     (if (<= rand 1.05e+73) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * sqrt(a));
	double tmp;
	if (rand <= -1.65e+47) {
		tmp = t_0;
	} else if (rand <= 1.05e+73) {
		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 = rand * (0.3333333333333333d0 * sqrt(a))
    if (rand <= (-1.65d+47)) then
        tmp = t_0
    else if (rand <= 1.05d+73) 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 = rand * (0.3333333333333333 * Math.sqrt(a));
	double tmp;
	if (rand <= -1.65e+47) {
		tmp = t_0;
	} else if (rand <= 1.05e+73) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (0.3333333333333333 * math.sqrt(a))
	tmp = 0
	if rand <= -1.65e+47:
		tmp = t_0
	elif rand <= 1.05e+73:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(0.3333333333333333 * sqrt(a)))
	tmp = 0.0
	if (rand <= -1.65e+47)
		tmp = t_0;
	elseif (rand <= 1.05e+73)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (0.3333333333333333 * sqrt(a));
	tmp = 0.0;
	if (rand <= -1.65e+47)
		tmp = t_0;
	elseif (rand <= 1.05e+73)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.65e+47], t$95$0, If[LessEqual[rand, 1.05e+73], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.65e47 or 1.0500000000000001e73 < rand

   1. Initial program 98.9%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   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-lft-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

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

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 96.4

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

   7. Applied rewrites 96.4%

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

   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 90.1%

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

   if -1.65e47 < rand < 1.0500000000000001e73

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 3: 91.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.3333333333333333 \cdot rand\right) \cdot \sqrt{a}\\ \mathbf{if}\;rand \leq -1.65 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.05 \cdot 10^{+73}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (* 0.3333333333333333 rand) (sqrt a))))
   (if (<= rand -1.65e+47)
     t_0
     (if (<= rand 1.05e+73) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = (0.3333333333333333 * rand) * sqrt(a);
	double tmp;
	if (rand <= -1.65e+47) {
		tmp = t_0;
	} else if (rand <= 1.05e+73) {
		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 <= (-1.65d+47)) then
        tmp = t_0
    else if (rand <= 1.05d+73) 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 <= -1.65e+47) {
		tmp = t_0;
	} else if (rand <= 1.05e+73) {
		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 <= -1.65e+47:
		tmp = t_0
	elif rand <= 1.05e+73:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(Float64(0.3333333333333333 * rand) * sqrt(a))
	tmp = 0.0
	if (rand <= -1.65e+47)
		tmp = t_0;
	elseif (rand <= 1.05e+73)
		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 <= -1.65e+47)
		tmp = t_0;
	elseif (rand <= 1.05e+73)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.65e+47], t$95$0, If[LessEqual[rand, 1.05e+73], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.65e47 or 1.0500000000000001e73 < rand

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft1-in N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in rand around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 93.0

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

   9. Applied rewrites 93.0%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 90.0%

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

   if -1.65e47 < rand < 1.0500000000000001e73

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.65 \cdot 10^{+47}:\\ \;\;\;\;\left(0.3333333333333333 \cdot rand\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 1.05 \cdot 10^{+73}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(0.3333333333333333 \cdot rand\right) \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 5: 99.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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-lft-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 98.6

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

7. Applied rewrites 98.6%

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

Alternative 6: 98.9% accurate, 2.7× speedup

Math

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

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft1-in N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 98.6

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

7. Applied rewrites 98.6%

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

Alternative 7: 99.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 98.5%

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

Alternative 8: 98.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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-lft-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

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

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 97.7

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

7. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right), a\right)} \]
8. Step-by-step derivation

9. Applied rewrites 97.8%

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

10. Final simplification 97.8%

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

Alternative 9: 98.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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-lft-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

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

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 97.7

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

7. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right), a\right)} \]
8. Step-by-step derivation

9. Applied rewrites 97.7%

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

Alternative 10: 98.1% accurate, 3.1× speedup

Math

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

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

Wolfram

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

Derivation

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. 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-lft-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.6%

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

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

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 97.7

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

7. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \sqrt{\frac{1}{a}} \cdot \left(0.3333333333333333 \cdot rand\right), a\right)} \]
8. Step-by-step derivation

9. Applied rewrites 97.7%

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

Alternative 11: 65.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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 62.6

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

5. Applied rewrites 62.6%

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

7. Applied rewrites 66.4%

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

Alternative 12: 62.7% 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 99.6%

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

3. Taylor expanded in rand around 0

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

   1. 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 62.6

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

5. Applied rewrites 62.6%

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

Alternative 13: 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 99.6%

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

3. Taylor expanded in rand around 0

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

   1. 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 62.6

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

5. Applied rewrites 62.6%

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

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 2024233 
(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))))