Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.7%

Time: 12.0s

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: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (* (+ a -0.3333333333333333) (+ 1.0 (/ rand (sqrt (fma 9.0 a -3.0))))))

C

double code(double a, double rand) {
	return (a + -0.3333333333333333) * (1.0 + (rand / sqrt(fma(9.0, a, -3.0))));
}

Julia

function code(a, rand)
	return Float64(Float64(a + -0.3333333333333333) * Float64(1.0 + Float64(rand / sqrt(fma(9.0, a, -3.0)))))
end

Wolfram

code[a_, rand_] := N[(N[(a + -0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(rand / N[Sqrt[N[(9.0 * a + -3.0), $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

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 99.8

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-+.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8

       \[\leadsto \left(a + \color{blue}{-0.3333333333333333}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(a + -0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}\right)} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.3 \cdot 10^{+70}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.3e+70)
   (* (* rand 0.3333333333333333) (sqrt a))
   (if (<= rand 2.4e+105)
     (+ a -0.3333333333333333)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.3e+70) {
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	} else if (rand <= 2.4e+105) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.3d+70)) then
        tmp = (rand * 0.3333333333333333d0) * sqrt(a)
    else if (rand <= 2.4d+105) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.3e+70) {
		tmp = (rand * 0.3333333333333333) * Math.sqrt(a);
	} else if (rand <= 2.4e+105) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.3e+70:
		tmp = (rand * 0.3333333333333333) * math.sqrt(a)
	elif rand <= 2.4e+105:
		tmp = a + -0.3333333333333333
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.3e+70)
		tmp = Float64(Float64(rand * 0.3333333333333333) * sqrt(a));
	elseif (rand <= 2.4e+105)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.3e+70)
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	elseif (rand <= 2.4e+105)
		tmp = a + -0.3333333333333333;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.3e+70], N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 2.4e+105], N[(a + -0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.3e70

   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

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lift--.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 87.7

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

   10. Applied rewrites 87.7%

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

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. pow-flip N/A

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

       6. pow1/2 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. div-inv N/A

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

       10. lift-/.f64 N/A

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

       11. *-commutative N/A

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

       12. *-commutative N/A

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

       13. lift-/.f64 N/A

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

       14. div-inv N/A

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

       15. associate-*l* N/A

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

       16. lift-sqrt.f64 N/A

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

       17. pow1/2 N/A

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

       18. pow-flip N/A

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

       19. pow-plus N/A

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

       20. metadata-eval N/A

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

       21. metadata-eval N/A

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

       22. pow1/2 N/A

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

       23. lift-sqrt.f64 N/A

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

   12. Applied rewrites 87.8%

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

   if -1.3e70 < rand < 2.39999999999999975e105

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 2.39999999999999975e105 < rand

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lift--.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 88.9

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

   10. Applied rewrites 88.9%

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

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. pow-flip N/A

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

       6. pow1/2 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. div-inv N/A

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

       10. lift-/.f64 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. lift-/.f64 N/A

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

       17. div-inv N/A

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

       18. associate-*l* N/A

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

       19. associate-/r/ N/A

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

       20. clear-num N/A

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

       21. lift-/.f64 N/A

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

       22. associate-*r* N/A

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

   12. Applied rewrites 89.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.3 \cdot 10^{+70}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{if}\;rand \leq -1.3 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (* rand 0.3333333333333333) (sqrt a))))
   (if (<= rand -1.3e+70)
     t_0
     (if (<= rand 2.4e+105) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = (rand * 0.3333333333333333) * sqrt(a);
	double tmp;
	if (rand <= -1.3e+70) {
		tmp = t_0;
	} else if (rand <= 2.4e+105) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (rand * 0.3333333333333333d0) * sqrt(a)
    if (rand <= (-1.3d+70)) then
        tmp = t_0
    else if (rand <= 2.4d+105) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = (rand * 0.3333333333333333) * Math.sqrt(a);
	double tmp;
	if (rand <= -1.3e+70) {
		tmp = t_0;
	} else if (rand <= 2.4e+105) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = (rand * 0.3333333333333333) * math.sqrt(a)
	tmp = 0
	if rand <= -1.3e+70:
		tmp = t_0
	elif rand <= 2.4e+105:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(Float64(rand * 0.3333333333333333) * sqrt(a))
	tmp = 0.0
	if (rand <= -1.3e+70)
		tmp = t_0;
	elseif (rand <= 2.4e+105)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = (rand * 0.3333333333333333) * sqrt(a);
	tmp = 0.0;
	if (rand <= -1.3e+70)
		tmp = t_0;
	elseif (rand <= 2.4e+105)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.3e+70], t$95$0, If[LessEqual[rand, 2.4e+105], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.3e70 or 2.39999999999999975e105 < rand

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lift--.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 96.6

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 88.2

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

   10. Applied rewrites 88.2%

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

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. pow-flip N/A

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

       6. pow1/2 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. associate-*l* N/A

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

       9. div-inv N/A

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

       10. lift-/.f64 N/A

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

       11. *-commutative N/A

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

       12. *-commutative N/A

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

       13. lift-/.f64 N/A

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

       14. div-inv N/A

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

       15. associate-*l* N/A

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

       16. lift-sqrt.f64 N/A

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

       17. pow1/2 N/A

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

       18. pow-flip N/A

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

       19. pow-plus N/A

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

       20. metadata-eval N/A

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

       21. metadata-eval N/A

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

       22. pow1/2 N/A

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

       23. lift-sqrt.f64 N/A

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

   12. Applied rewrites 88.2%

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

   if -1.3e70 < rand < 2.39999999999999975e105

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.3 \cdot 10^{+70}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.3 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* 0.3333333333333333 (sqrt a)))))
   (if (<= rand -1.3e+70)
     t_0
     (if (<= rand 2.4e+105) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * sqrt(a));
	double tmp;
	if (rand <= -1.3e+70) {
		tmp = t_0;
	} else if (rand <= 2.4e+105) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = rand * (0.3333333333333333d0 * sqrt(a))
    if (rand <= (-1.3d+70)) then
        tmp = t_0
    else if (rand <= 2.4d+105) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * Math.sqrt(a));
	double tmp;
	if (rand <= -1.3e+70) {
		tmp = t_0;
	} else if (rand <= 2.4e+105) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (0.3333333333333333 * math.sqrt(a))
	tmp = 0
	if rand <= -1.3e+70:
		tmp = t_0
	elif rand <= 2.4e+105:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(0.3333333333333333 * sqrt(a)))
	tmp = 0.0
	if (rand <= -1.3e+70)
		tmp = t_0;
	elseif (rand <= 2.4e+105)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (0.3333333333333333 * sqrt(a));
	tmp = 0.0;
	if (rand <= -1.3e+70)
		tmp = t_0;
	elseif (rand <= 2.4e+105)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.3e+70], t$95$0, If[LessEqual[rand, 2.4e+105], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.3e70 or 2.39999999999999975e105 < rand

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lift--.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 96.6

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 88.2

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

   10. Applied rewrites 88.2%

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

   if -1.3e70 < rand < 2.39999999999999975e105

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

Alternative 5: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 6: 68.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a + -0.3333333333333333\right) \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 4.1e+154)
   (+ a -0.3333333333333333)
   (/ (* (+ a -0.3333333333333333) rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 4.1d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = ((a + (-0.3333333333333333d0)) * rand) / rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 4.1e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 4.1e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = ((a + -0.3333333333333333) * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 4.1e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(a + -0.3333333333333333) * rand) / rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 4.1e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 4.1e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(N[(a + -0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision] / rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 4.1e154

   1. Initial program 99.8%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 69.0

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

   5. Applied rewrites 69.0%

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

   if 4.1e154 < rand

   1. Initial program 99.8%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. 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

      1. lift-+.f64 5.8

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{3}\right) \cdot \color{blue}{{rand}^{0}} \]

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. inv-pow N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 46.9

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

   7. Applied rewrites 46.9%

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

Alternative 7: 98.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in a around inf

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

   1. lower-sqrt.f64 98.6

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

8. Applied rewrites 98.6%

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

9. Final simplification 98.6%

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

Alternative 8: 97.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. *-rgt-identity N/A

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

   11. lift--.f64 N/A

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

   12. associate-+l- N/A

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

   13. lower--.f64 N/A

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

4. Applied rewrites 87.6%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 97.8

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

7. Applied rewrites 97.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 97.8

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

9. Applied rewrites 97.8%

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

10. Final simplification 97.8%

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

Alternative 9: 66.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 3.3 \cdot 10^{+141}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 3.3e+141) (+ a -0.3333333333333333) (* 3.0 (* a a))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 3.3e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 3.0 * (a * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 3.3d+141) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 3.0d0 * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 3.3e+141) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 3.0 * (a * a);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 3.3e+141:
		tmp = a + -0.3333333333333333
	else:
		tmp = 3.0 * (a * a)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 3.3e+141)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(3.0 * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 3.3e+141)
		tmp = a + -0.3333333333333333;
	else
		tmp = 3.0 * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 3.3e+141], N[(a + -0.3333333333333333), $MachinePrecision], N[(3.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 3.2999999999999997e141

   1. Initial program 99.8%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if 3.2999999999999997e141 < rand

   1. Initial program 99.7%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

      1. flip-+ N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 34.2

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

   7. Applied rewrites 34.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 35.1%

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

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{3 \cdot {a}^{2}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{3 \cdot {a}^{2}} \]

       2. unpow2 N/A

          \[\leadsto 3 \cdot \color{blue}{\left(a \cdot a\right)} \]

       3. lower-*.f64 35.1

          \[\leadsto 3 \cdot \color{blue}{\left(a \cdot a\right)} \]

   13. Applied rewrites 35.1%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.7% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 62.6

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

5. Applied rewrites 62.6%

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

Alternative 11: 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. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 62.6

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

5. Applied rewrites 62.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 1.7%

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

Reproduce

herbie shell --seed 2024214 
(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))))