Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 9.2s

Alternatives: 11

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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / 3.0), $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. 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

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

4. Applied rewrites 99.9%

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

Alternative 2: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.36 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{elif}\;rand \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.36e+52)
   (* (* (sqrt a) 0.3333333333333333) rand)
   (if (<= rand 1.5e+82)
     (- a 0.3333333333333333)
     (* (* (sqrt a) rand) 0.3333333333333333))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.36e+52) {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	} else if (rand <= 1.5e+82) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.36d+52)) then
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    else if (rand <= 1.5d+82) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = (sqrt(a) * rand) * 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.36e+52) {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	} else if (rand <= 1.5e+82) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = (Math.sqrt(a) * rand) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.36e+52:
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	elif rand <= 1.5e+82:
		tmp = a - 0.3333333333333333
	else:
		tmp = (math.sqrt(a) * rand) * 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.36e+52)
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	elseif (rand <= 1.5e+82)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(Float64(sqrt(a) * rand) * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.36e+52)
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	elseif (rand <= 1.5e+82)
		tmp = a - 0.3333333333333333;
	else
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.36e+52], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], If[LessEqual[rand, 1.5e+82], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.35999999999999994e52

   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. 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 99.8%

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

   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. lft-mult-inverse N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. rgt-mult-inverse N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 81.9%

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

   12. Applied rewrites 82.0%

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

   if -1.35999999999999994e52 < rand < 1.49999999999999995e82

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 1.49999999999999995e82 < rand

   1. Initial program 97.7%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing
   3. 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 97.9%

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

   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. lft-mult-inverse N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. rgt-mult-inverse N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 94.4

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 88.7%

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

Alternative 3: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.36 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{elif}\;rand \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.36e+52)
   (* (* (sqrt a) 0.3333333333333333) rand)
   (if (<= rand 1.5e+82)
     (- a 0.3333333333333333)
     (* (sqrt a) (* rand 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.36e+52) {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	} else if (rand <= 1.5e+82) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = sqrt(a) * (rand * 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.36d+52)) then
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    else if (rand <= 1.5d+82) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = sqrt(a) * (rand * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.36e+52) {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	} else if (rand <= 1.5e+82) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = Math.sqrt(a) * (rand * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.36e+52:
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	elif rand <= 1.5e+82:
		tmp = a - 0.3333333333333333
	else:
		tmp = math.sqrt(a) * (rand * 0.3333333333333333)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.36e+52)
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	elseif (rand <= 1.5e+82)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(sqrt(a) * Float64(rand * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.36e+52)
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	elseif (rand <= 1.5e+82)
		tmp = a - 0.3333333333333333;
	else
		tmp = sqrt(a) * (rand * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.36e+52], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], If[LessEqual[rand, 1.5e+82], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[Sqrt[a], $MachinePrecision] * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.35999999999999994e52

   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. 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 99.8%

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

   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. lft-mult-inverse N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. rgt-mult-inverse N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 81.9%

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

   12. Applied rewrites 82.0%

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

   if -1.35999999999999994e52 < rand < 1.49999999999999995e82

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 1.49999999999999995e82 < rand

   1. Initial program 97.7%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing
   3. 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 97.9%

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

   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. lft-mult-inverse N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. rgt-mult-inverse N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 94.4

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 88.7%

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

   12. Applied rewrites 88.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.36 \cdot 10^{+52}:\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{elif}\;rand \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -1.36 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt a) (* rand 0.3333333333333333))))
   (if (<= rand -1.36e+52)
     t_0
     (if (<= rand 1.5e+82) (- a 0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt(a) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -1.36e+52) {
		tmp = t_0;
	} else if (rand <= 1.5e+82) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(a) * (rand * 0.3333333333333333d0)
    if (rand <= (-1.36d+52)) then
        tmp = t_0
    else if (rand <= 1.5d+82) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt(a) * (rand * 0.3333333333333333);
	double tmp;
	if (rand <= -1.36e+52) {
		tmp = t_0;
	} else if (rand <= 1.5e+82) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt(a) * (rand * 0.3333333333333333)
	tmp = 0
	if rand <= -1.36e+52:
		tmp = t_0
	elif rand <= 1.5e+82:
		tmp = a - 0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(a) * Float64(rand * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -1.36e+52)
		tmp = t_0;
	elseif (rand <= 1.5e+82)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt(a) * (rand * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -1.36e+52)
		tmp = t_0;
	elseif (rand <= 1.5e+82)
		tmp = a - 0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[a], $MachinePrecision] * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.36e+52], t$95$0, If[LessEqual[rand, 1.5e+82], N[(a - 0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.35999999999999994e52 or 1.49999999999999995e82 < rand

   1. Initial program 99.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Add Preprocessing
   3. 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 99.0%

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

   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. lft-mult-inverse N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. rgt-mult-inverse N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 95.5

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 84.8%

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

   12. Applied rewrites 84.8%

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

   if -1.35999999999999994e52 < rand < 1.49999999999999995e82

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.36 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 1.5 \cdot 10^{+82}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
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. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

Alternative 6: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[N[(a - 0.3333333333333333), $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. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 7: 98.8% accurate, 2.7× speedup

Math

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

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

Wolfram

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

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 98.2

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

7. Applied rewrites 98.2%

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

Alternative 8: 67.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 2.05e+142)
   (- a 0.3333333333333333)
   (/ (fma a a -0.1111111111111111) 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 2.05e+142) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = fma(a, a, -0.1111111111111111) / 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 2.05e+142)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(fma(a, a, -0.1111111111111111) / 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 2.05e+142], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[(a * a + -0.1111111111111111), $MachinePrecision] / 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 2.04999999999999991e142

   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. lower--.f64 69.4

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

   5. Applied rewrites 69.4%

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

   if 2.04999999999999991e142 < rand

   1. Initial program 97.2%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 37.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 38.6%

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

Alternative 9: 97.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

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. lft-mult-inverse N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. rgt-mult-inverse N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-/.f64 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 96.7%

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

Alternative 10: 63.6% 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. lower--.f64 61.5

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

5. Applied rewrites 61.5%

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

Alternative 11: 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. lower--.f64 61.5

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

5. Applied rewrites 61.5%

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

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 2024277 
(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))))