Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 13.1s

Alternatives: 14

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(N[(a + -0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + 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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. accelerator-lowering-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 egg-rr 99.8%

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

Alternative 2: 91.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -8.8 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -8.8e+57)
   (* (sqrt a) (* rand 0.3333333333333333))
   (if (<= rand 4.7e+95)
     (+ a -0.3333333333333333)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -8.8e+57) {
		tmp = sqrt(a) * (rand * 0.3333333333333333);
	} else if (rand <= 4.7e+95) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-8.8d+57)) then
        tmp = sqrt(a) * (rand * 0.3333333333333333d0)
    else if (rand <= 4.7d+95) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -8.8e+57) {
		tmp = Math.sqrt(a) * (rand * 0.3333333333333333);
	} else if (rand <= 4.7e+95) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -8.8e+57:
		tmp = math.sqrt(a) * (rand * 0.3333333333333333)
	elif rand <= 4.7e+95:
		tmp = a + -0.3333333333333333
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -8.8e+57)
		tmp = Float64(sqrt(a) * Float64(rand * 0.3333333333333333));
	elseif (rand <= 4.7e+95)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -8.8e+57)
		tmp = sqrt(a) * (rand * 0.3333333333333333);
	elseif (rand <= 4.7e+95)
		tmp = a + -0.3333333333333333;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -8.8e+57], N[(N[Sqrt[a], $MachinePrecision] * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.7e+95], N[(a + -0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -8.8000000000000003e57

   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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-lowering-*.f64 88.2

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

   5. Simplified 88.2%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-lowering-sqrt.f64 86.4

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

   8. Simplified 86.4%

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

   if -8.8000000000000003e57 < rand < 4.69999999999999972e95

   1. Initial program 99.9%

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

   3. 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. +-lowering-+.f64 93.5

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

   5. Simplified 93.5%

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

   if 4.69999999999999972e95 < 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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-lowering-*.f64 96.1

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

   5. Simplified 96.1%

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

   6. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 96.2

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

   8. Simplified 96.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8.8 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -8.8 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -8.8e+57)
     t_0
     (if (<= rand 4.7e+95) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -8.8e+57) {
		tmp = t_0;
	} else if (rand <= 4.7e+95) {
		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 <= (-8.8d+57)) then
        tmp = t_0
    else if (rand <= 4.7d+95) 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 <= -8.8e+57) {
		tmp = t_0;
	} else if (rand <= 4.7e+95) {
		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 <= -8.8e+57:
		tmp = t_0
	elif rand <= 4.7e+95:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.3333333333333333 * Float64(rand * sqrt(a)))
	tmp = 0.0
	if (rand <= -8.8e+57)
		tmp = t_0;
	elseif (rand <= 4.7e+95)
		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 <= -8.8e+57)
		tmp = t_0;
	elseif (rand <= 4.7e+95)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.8e+57], t$95$0, If[LessEqual[rand, 4.7e+95], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -8.8000000000000003e57 or 4.69999999999999972e95 < 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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-lowering-*.f64 92.3

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

   5. Simplified 92.3%

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

   6. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 90.7

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

   8. Simplified 90.7%

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

   if -8.8000000000000003e57 < rand < 4.69999999999999972e95

   1. Initial program 99.9%

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

   3. 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. +-lowering-+.f64 93.5

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

   5. Simplified 93.5%

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

Alternative 4: 99.8% accurate, 2.4× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. accelerator-lowering-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. sqrt-lowering-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. +-lowering-+.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. *-lowering-*.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. +-lowering-+.f64 99.8

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 68.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 2.1 \cdot 10^{+157}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a + -0.3333333333333333\right) \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 2.1e+157)
   (+ a -0.3333333333333333)
   (/ (* (+ a -0.3333333333333333) rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 2.1e+157) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 2.1d+157) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = ((a + (-0.3333333333333333d0)) * rand) / rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 2.1e+157) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 2.1e+157:
		tmp = a + -0.3333333333333333
	else:
		tmp = ((a + -0.3333333333333333) * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 2.1e+157)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(a + -0.3333333333333333) * rand) / rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 2.1e+157)
		tmp = a + -0.3333333333333333;
	else
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 2.1e+157], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(N[(a + -0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision] / rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 2.1e157

   1. Initial program 99.9%

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

   3. 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. +-lowering-+.f64 73.8

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

   5. Simplified 73.8%

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

   if 2.1e157 < 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 inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in rand around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 6.1

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

   8. Simplified 6.1%

      \[\leadsto rand \cdot \color{blue}{\frac{-0.3333333333333333 + a}{rand}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 52.0

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

   10. Applied egg-rr 52.0%

       \[\leadsto \color{blue}{\frac{rand \cdot \left(a + -0.3333333333333333\right)}{rand}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 2.1 \cdot 10^{+157}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a + -0.3333333333333333\right) \cdot rand}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% 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 99.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. accelerator-lowering-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 egg-rr 99.8%

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

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. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 99.3

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

7. Simplified 99.3%

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

Alternative 7: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. sqrt-prod N/A

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

   7. metadata-eval N/A

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

   8. times-frac N/A

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

   9. *-rgt-identity N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in a around inf

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

   1. sqrt-lowering-sqrt.f64 99.3

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

7. Simplified 99.3%

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

Alternative 8: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. sqrt-prod N/A

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

   7. metadata-eval N/A

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

   8. times-frac N/A

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

   9. *-rgt-identity N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in a around inf

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

   1. sqrt-lowering-sqrt.f64 99.3

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

7. Simplified 99.3%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. +-lowering-+.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-lowering-*.f64 99.3

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

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

Alternative 9: 67.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 1.95e+149)
   (+ a -0.3333333333333333)
   (/ (fma a a -0.1111111111111111) 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 1.95e+149) {
		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 <= 1.95e+149)
		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, 1.95e+149], 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 < 1.95e149

   1. Initial program 99.9%

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

   3. 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. +-lowering-+.f64 73.8

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

   5. Simplified 73.8%

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

   if 1.95e149 < 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. 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. +-lowering-+.f64 6.1

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

   5. Simplified 6.1%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 41.9

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

   7. Applied egg-rr 41.9%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 42.5%

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

Alternative 10: 97.9% 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 99.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. accelerator-lowering-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 egg-rr 99.8%

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

5. Taylor expanded in a around inf

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

7. Simplified 99.2%

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

8. Taylor expanded in a around inf

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 99.0

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

10. Simplified 99.0%

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

Alternative 11: 97.9% accurate, 3.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. sqrt-prod N/A

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

   7. metadata-eval N/A

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

   8. times-frac N/A

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

   9. *-rgt-identity N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in a around inf

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

   1. sqrt-lowering-sqrt.f64 99.3

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

7. Simplified 99.3%

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

8. Taylor expanded in a around inf

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

10. Simplified 98.9%

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

Alternative 12: 62.9% 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. 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. +-lowering-+.f64 63.5

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

5. Simplified 63.5%

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

Alternative 13: 61.9% accurate, 68.0× speedup

Math

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

FPCore

(FPCore (a rand) :precision binary64 a)

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return a

Julia

function code(a, rand)
	return a
end

MATLAB

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

Wolfram

code[a_, rand_] := a

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. 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. +-lowering-+.f64 63.5

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

5. Simplified 63.5%

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

6. Taylor expanded in a around inf

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

8. Simplified 63.2%

   \[\leadsto \color{blue}{a} \]
9. Add Preprocessing

Alternative 14: 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. 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. +-lowering-+.f64 63.5

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

5. Simplified 63.5%

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

6. Taylor expanded in a around 0

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

8. Simplified 1.5%

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

Reproduce

herbie shell --seed 2024194 
(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))))