Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.7%

Time: 12.1s

Alternatives: 9

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

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*l/ N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-neg N/A

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

   15. distribute-lft-in N/A

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

   16. lift-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

4. Applied egg-rr 99.9%

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

   1. metadata-eval N/A

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

   2. lower--.f64 99.9

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

6. Applied egg-rr 99.9%

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

Alternative 2: 90.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1.3 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -1.3e+45)
     t_0
     (if (<= rand 6.5e+76) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -1.3e+45) {
		tmp = t_0;
	} else if (rand <= 6.5e+76) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (rand * sqrt(a))
    if (rand <= (-1.3d+45)) then
        tmp = t_0
    else if (rand <= 6.5d+76) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * Math.sqrt(a));
	double tmp;
	if (rand <= -1.3e+45) {
		tmp = t_0;
	} else if (rand <= 6.5e+76) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.3333333333333333 * (rand * math.sqrt(a))
	tmp = 0
	if rand <= -1.3e+45:
		tmp = t_0
	elif rand <= 6.5e+76:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.3333333333333333 * Float64(rand * sqrt(a)))
	tmp = 0.0
	if (rand <= -1.3e+45)
		tmp = t_0;
	elseif (rand <= 6.5e+76)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.3333333333333333 * (rand * sqrt(a));
	tmp = 0.0;
	if (rand <= -1.3e+45)
		tmp = t_0;
	elseif (rand <= 6.5e+76)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.3e+45], t$95$0, If[LessEqual[rand, 6.5e+76], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.30000000000000004e45 or 6.5000000000000005e76 < 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

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-sqrt.f64 99.6

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. lower-+.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval 99.6

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

   4. Applied egg-rr 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around inf

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

      1. lower-sqrt.f64 98.6

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

   9. Simplified 98.6%

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

   10. Taylor expanded in rand around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 89.9

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

   12. Simplified 89.9%

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

   if -1.30000000000000004e45 < rand < 6.5000000000000005e76

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 100.0

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. distribute-lft-in N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. 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 96.9

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

   7. Simplified 96.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.3 \cdot 10^{+45}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f64 99.8

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. lower-+.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 99.9

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

8. Applied egg-rr 99.9%

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

Alternative 4: 69.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 4.2 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{rand \cdot \left(a + -0.3333333333333333\right)}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 4.2e+154)
   (+ a -0.3333333333333333)
   (/ (* rand (+ a -0.3333333333333333)) rand)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 4.2e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (rand * (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 <= 4.2d+154) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (rand * (a + (-0.3333333333333333d0))) / rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 4.2e+154) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (rand * (a + -0.3333333333333333)) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 4.2e+154:
		tmp = a + -0.3333333333333333
	else:
		tmp = (rand * (a + -0.3333333333333333)) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 4.2e+154)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(rand * Float64(a + -0.3333333333333333)) / rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 4.2e+154)
		tmp = a + -0.3333333333333333;
	else
		tmp = (rand * (a + -0.3333333333333333)) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 4.2e+154], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(rand * N[(a + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 4.19999999999999989e154

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 99.9

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. distribute-lft-in N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. 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 72.2

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

   7. Simplified 72.2%

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

   if 4.19999999999999989e154 < 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

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 99.8

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. distribute-lft-in N/A

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

      16. lift-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in rand around 0

      \[\leadsto \color{blue}{a - \frac{1}{3}} \]
   6. 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 5.5

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

   7. Simplified 5.5%

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

      1. lift-+.f64 5.5

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

      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. lower-/.f64 N/A

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

      11. lower-*.f64 54.0

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

   9. Applied egg-rr 54.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 4.2 \cdot 10^{+154}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{rand \cdot \left(a + -0.3333333333333333\right)}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-sqrt.f64 99.8

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. lower-+.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in a around inf

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

   1. lower-sqrt.f64 99.1

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 6: 97.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*l/ N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-neg N/A

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

   15. distribute-lft-in N/A

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

   16. lift-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.0

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

7. Simplified 98.0%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. lift-/.f64 N/A

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

   10. inv-pow N/A

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

   11. pow-pow N/A

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

   12. pow-plus N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow1/2 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*l* N/A

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

   19. *-commutative N/A

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

   20. *-lft-identity N/A

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

9. Applied egg-rr 98.0%

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

Alternative 7: 97.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*l/ N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-neg N/A

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

   15. distribute-lft-in N/A

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

   16. lift-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.0

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

7. Simplified 98.0%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-/.f64 N/A

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

   11. inv-pow N/A

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

   12. pow-pow N/A

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

   13. pow-plus N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. pow1/2 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lower-fma.f64 98.0

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

9. Applied egg-rr 98.0%

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

Alternative 8: 63.2% 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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*l/ N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-neg N/A

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

   15. distribute-lft-in N/A

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

   16. lift-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
6. 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 62.3

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

7. Simplified 62.3%

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

Alternative 9: 1.5% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*l/ N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-neg N/A

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

   15. distribute-lft-in N/A

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

   16. lift-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in rand around 0

   \[\leadsto \color{blue}{a - \frac{1}{3}} \]
6. 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 62.3

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

7. Simplified 62.3%

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

8. Taylor expanded in a around 0

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

10. Simplified 1.5%

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

Reproduce

herbie shell --seed 2024215 
(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))))