Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.9%

Time: 8.9s

Alternatives: 12

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = a - (0.3333333333333333d0 - ((rand / 3.0d0) * sqrt((a - 0.3333333333333333d0))))
end function

Java

public static double code(double a, double rand) {
	return a - (0.3333333333333333 - ((rand / 3.0) * Math.sqrt((a - 0.3333333333333333))));
}

Python

def code(a, rand):
	return a - (0.3333333333333333 - ((rand / 3.0) * math.sqrt((a - 0.3333333333333333))))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = a - (0.3333333333333333 - ((rand / 3.0) * sqrt((a - 0.3333333333333333))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 99.8

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. distribute-rgt-in N/A

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

   14. sub-neg N/A

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

   15. lift--.f64 N/A

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

   16. sqrt-prod N/A

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

   17. metadata-eval N/A

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

   18. lift-sqrt.f64 N/A

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

   19. times-frac N/A

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

   20. clear-num N/A

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

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{a - \left(0.3333333333333333 - \frac{rand}{3} \cdot \sqrt{a - 0.3333333333333333}\right)} \]
7. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - {3}^{-1} \leq 1000000000000:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;a - \left(\sqrt{a} \cdot rand\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= (- a (pow 3.0 -1.0)) 1000000000000.0)
   (- a 0.3333333333333333)
   (- a (* (* (sqrt a) rand) -0.3333333333333333))))

C

double code(double a, double rand) {
	double tmp;
	if ((a - pow(3.0, -1.0)) <= 1000000000000.0) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = a - ((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 ((a - (3.0d0 ** (-1.0d0))) <= 1000000000000.0d0) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = a - ((sqrt(a) * rand) * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((a - Math.pow(3.0, -1.0)) <= 1000000000000.0) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = a - ((Math.sqrt(a) * rand) * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (a - math.pow(3.0, -1.0)) <= 1000000000000.0:
		tmp = a - 0.3333333333333333
	else:
		tmp = a - ((math.sqrt(a) * rand) * -0.3333333333333333)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (Float64(a - (3.0 ^ -1.0)) <= 1000000000000.0)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(a - Float64(Float64(sqrt(a) * rand) * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((a - (3.0 ^ -1.0)) <= 1000000000000.0)
		tmp = a - 0.3333333333333333;
	else
		tmp = a - ((sqrt(a) * rand) * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[N[(a - N[Power[3.0, -1.0], $MachinePrecision]), $MachinePrecision], 1000000000000.0], N[(a - 0.3333333333333333), $MachinePrecision], N[(a - N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a (/.f64 #s(literal 1 binary64) #s(literal 3 binary64))) < 1e12

   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 82.0

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

   5. Applied rewrites 82.0%

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

   if 1e12 < (-.f64 a (/.f64 #s(literal 1 binary64) #s(literal 3 binary64)))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 99.8

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. sub-neg N/A

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

      15. lift--.f64 N/A

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

      16. sqrt-prod N/A

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

      17. metadata-eval N/A

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

      18. lift-sqrt.f64 N/A

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

      19. times-frac N/A

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

      20. clear-num N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - {3}^{-1} \leq 1000000000000:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;a - \left(\sqrt{a} \cdot rand\right) \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -5.5 \cdot 10^{+116} \lor \neg \left(rand \leq 2.8 \cdot 10^{+94}\right):\\ \;\;\;\;\left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -5.5e+116) (not (<= rand 2.8e+94)))
   (* (* (sqrt (- a 0.3333333333333333)) 0.3333333333333333) rand)
   (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -5.5e+116) || !(rand <= 2.8e+94)) {
		tmp = (sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand;
	} else {
		tmp = a - 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 <= (-5.5d+116)) .or. (.not. (rand <= 2.8d+94))) then
        tmp = (sqrt((a - 0.3333333333333333d0)) * 0.3333333333333333d0) * rand
    else
        tmp = a - 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -5.5e+116) || !(rand <= 2.8e+94)) {
		tmp = (Math.sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -5.5e+116) or not (rand <= 2.8e+94):
		tmp = (math.sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand
	else:
		tmp = a - 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -5.5e+116) || !(rand <= 2.8e+94))
		tmp = Float64(Float64(sqrt(Float64(a - 0.3333333333333333)) * 0.3333333333333333) * rand);
	else
		tmp = Float64(a - 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -5.5e+116) || ~((rand <= 2.8e+94)))
		tmp = (sqrt((a - 0.3333333333333333)) * 0.3333333333333333) * rand;
	else
		tmp = a - 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -5.5e+116], N[Not[LessEqual[rand, 2.8e+94]], $MachinePrecision]], N[(N[(N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], N[(a - 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -5.50000000000000035e116 or 2.79999999999999998e94 < rand

   1. Initial program 99.5%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 91.8%

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

   if -5.50000000000000035e116 < rand < 2.79999999999999998e94

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -5.5 \cdot 10^{+116} \lor \neg \left(rand \leq 2.8 \cdot 10^{+94}\right):\\ \;\;\;\;\left(\sqrt{a - 0.3333333333333333} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a - 0.3333333333333333}\\ \mathbf{if}\;rand \leq -5.5 \cdot 10^{+116}:\\ \;\;\;\;\left(0.3333333333333333 \cdot rand\right) \cdot t\_0\\ \mathbf{elif}\;rand \leq 2.8 \cdot 10^{+94}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 0.3333333333333333\right) \cdot rand\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (sqrt (- a 0.3333333333333333))))
   (if (<= rand -5.5e+116)
     (* (* 0.3333333333333333 rand) t_0)
     (if (<= rand 2.8e+94)
       (- a 0.3333333333333333)
       (* (* t_0 0.3333333333333333) rand)))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a - 0.3333333333333333));
	double tmp;
	if (rand <= -5.5e+116) {
		tmp = (0.3333333333333333 * rand) * t_0;
	} else if (rand <= 2.8e+94) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = (t_0 * 0.3333333333333333) * rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a - 0.3333333333333333d0))
    if (rand <= (-5.5d+116)) then
        tmp = (0.3333333333333333d0 * rand) * t_0
    else if (rand <= 2.8d+94) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = (t_0 * 0.3333333333333333d0) * rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a - 0.3333333333333333));
	double tmp;
	if (rand <= -5.5e+116) {
		tmp = (0.3333333333333333 * rand) * t_0;
	} else if (rand <= 2.8e+94) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = (t_0 * 0.3333333333333333) * rand;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a - 0.3333333333333333))
	tmp = 0
	if rand <= -5.5e+116:
		tmp = (0.3333333333333333 * rand) * t_0
	elif rand <= 2.8e+94:
		tmp = a - 0.3333333333333333
	else:
		tmp = (t_0 * 0.3333333333333333) * rand
	return tmp

Julia

function code(a, rand)
	t_0 = sqrt(Float64(a - 0.3333333333333333))
	tmp = 0.0
	if (rand <= -5.5e+116)
		tmp = Float64(Float64(0.3333333333333333 * rand) * t_0);
	elseif (rand <= 2.8e+94)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(Float64(t_0 * 0.3333333333333333) * rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a - 0.3333333333333333));
	tmp = 0.0;
	if (rand <= -5.5e+116)
		tmp = (0.3333333333333333 * rand) * t_0;
	elseif (rand <= 2.8e+94)
		tmp = a - 0.3333333333333333;
	else
		tmp = (t_0 * 0.3333333333333333) * rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[rand, -5.5e+116], N[(N[(0.3333333333333333 * rand), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[rand, 2.8e+94], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[(t$95$0 * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -5.50000000000000035e116

   1. Initial program 99.5%

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

   3. 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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -5.50000000000000035e116 < rand < 2.79999999999999998e94

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 2.79999999999999998e94 < rand

   1. Initial program 99.4%

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

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 89.8%

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

Alternative 5: 99.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 99.8

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. *-lft-identity N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. sqr-pow N/A

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

   6. sqr-pow N/A

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

   7. lift-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. sub-neg N/A

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

   12. pow1/2 N/A

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

   13. associate-*l/ N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

6. Applied rewrites 99.9%

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

Alternative 6: 91.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -5.5 \cdot 10^{+116} \lor \neg \left(rand \leq 3.1 \cdot 10^{+94}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -5.5e+116) (not (<= rand 3.1e+94)))
   (* (* (sqrt a) 0.3333333333333333) rand)
   (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -5.5e+116) || !(rand <= 3.1e+94)) {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = a - 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 <= (-5.5d+116)) .or. (.not. (rand <= 3.1d+94))) then
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    else
        tmp = a - 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -5.5e+116) || !(rand <= 3.1e+94)) {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = a - 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -5.5e+116) or not (rand <= 3.1e+94):
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	else:
		tmp = a - 0.3333333333333333
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -5.5e+116) || !(rand <= 3.1e+94))
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	else
		tmp = Float64(a - 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -5.5e+116) || ~((rand <= 3.1e+94)))
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	else
		tmp = a - 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -5.5e+116], N[Not[LessEqual[rand, 3.1e+94]], $MachinePrecision]], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], N[(a - 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -5.50000000000000035e116 or 3.09999999999999991e94 < rand

   1. Initial program 99.5%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 90.8%

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

   if -5.50000000000000035e116 < rand < 3.09999999999999991e94

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -5.5 \cdot 10^{+116} \lor \neg \left(rand \leq 3.1 \cdot 10^{+94}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;a - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -5.5 \cdot 10^{+116}:\\ \;\;\;\;\left(0.3333333333333333 \cdot rand\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 3.1 \cdot 10^{+94}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -5.5e+116)
   (* (* 0.3333333333333333 rand) (sqrt a))
   (if (<= rand 3.1e+94)
     (- a 0.3333333333333333)
     (* (* (sqrt a) 0.3333333333333333) rand))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -5.5e+116) {
		tmp = (0.3333333333333333 * rand) * sqrt(a);
	} else if (rand <= 3.1e+94) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = (sqrt(a) * 0.3333333333333333) * 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 <= (-5.5d+116)) then
        tmp = (0.3333333333333333d0 * rand) * sqrt(a)
    else if (rand <= 3.1d+94) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -5.5e+116) {
		tmp = (0.3333333333333333 * rand) * Math.sqrt(a);
	} else if (rand <= 3.1e+94) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -5.5e+116:
		tmp = (0.3333333333333333 * rand) * math.sqrt(a)
	elif rand <= 3.1e+94:
		tmp = a - 0.3333333333333333
	else:
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -5.5e+116)
		tmp = Float64(Float64(0.3333333333333333 * rand) * sqrt(a));
	elseif (rand <= 3.1e+94)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -5.5e+116)
		tmp = (0.3333333333333333 * rand) * sqrt(a);
	elseif (rand <= 3.1e+94)
		tmp = a - 0.3333333333333333;
	else
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -5.5e+116], N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 3.1e+94], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -5.50000000000000035e116

   1. Initial program 99.5%

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

   3. 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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in a around -inf

      \[\leadsto \left(\frac{1}{3} \cdot rand\right) \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{a} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 92.3%

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

   if -5.50000000000000035e116 < rand < 3.09999999999999991e94

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. lower--.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 3.09999999999999991e94 < rand

   1. Initial program 99.4%

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

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 89.7%

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

Alternative 8: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 9: 98.8% accurate, 2.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 99.8

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+l- N/A

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

   5. lower--.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. distribute-rgt-in N/A

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

   14. sub-neg N/A

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

   15. lift--.f64 N/A

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

   16. sqrt-prod N/A

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

   17. metadata-eval N/A

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

   18. lift-sqrt.f64 N/A

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

   19. times-frac N/A

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

   20. clear-num N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in a around inf

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

   1. lower-sqrt.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. lift--.f64 N/A

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

    2. sub-neg N/A

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

    3. +-commutative N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. associate-*l/ N/A

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

    7. distribute-neg-frac2 N/A

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

    8. div-inv N/A

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

    9. metadata-eval N/A

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. lower-fma.f64 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f64 N/A

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

    15. metadata-eval 99.3

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

11. Applied rewrites 99.3%

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

Alternative 10: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 99.3%

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

Alternative 11: 62.1% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ a - 0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 (- a 0.3333333333333333))

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return a - 0.3333333333333333

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. lower--.f64 67.6

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

5. Applied rewrites 67.6%

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

Alternative 12: 1.6% accurate, 68.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a rand) :precision binary64 -0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. lower--.f64 67.6

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

5. Applied rewrites 67.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 1.4%

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

Reproduce

herbie shell --seed 2024306 
(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))))