Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 13

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. +-commutative N/A

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

   10. distribute-lft-in N/A

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(a + -0.3333333333333333), $MachinePrecision] * N[(N[(rand / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $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 egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 91.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -4.2 \cdot 10^{+82}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.9 \cdot 10^{+40}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -4.2e+82)
   (* 0.3333333333333333 (* rand (sqrt a)))
   (if (<= rand 1.9e+40)
     (+ a -0.3333333333333333)
     (* rand (* 0.3333333333333333 (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -4.2e+82) {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	} else if (rand <= 1.9e+40) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = rand * (0.3333333333333333 * 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 <= (-4.2d+82)) then
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    else if (rand <= 1.9d+40) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = rand * (0.3333333333333333d0 * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -4.2e+82) {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	} else if (rand <= 1.9e+40) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = rand * (0.3333333333333333 * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -4.2e+82:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	elif rand <= 1.9e+40:
		tmp = a + -0.3333333333333333
	else:
		tmp = rand * (0.3333333333333333 * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -4.2e+82)
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	elseif (rand <= 1.9e+40)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(rand * Float64(0.3333333333333333 * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -4.2e+82)
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	elseif (rand <= 1.9e+40)
		tmp = a + -0.3333333333333333;
	else
		tmp = rand * (0.3333333333333333 * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -4.2e+82], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.9e+40], N[(a + -0.3333333333333333), $MachinePrecision], N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -4.2e82

   1. Initial program 99.5%

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

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

      10. distribute-lft-in N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 97.6

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

   7. Simplified 97.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 97.6

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

   9. Applied egg-rr 97.6%

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

   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 92.1

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

   12. Simplified 92.1%

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

   if -4.2e82 < rand < 1.90000000000000002e40

   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 95.8

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

   5. Simplified 95.8%

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

   if 1.90000000000000002e40 < rand

   1. Initial program 99.6%

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

   3. Taylor expanded in rand around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in rand around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 82.5

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

   8. Simplified 82.5%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)} \]
   10. 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 80.9

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

   11. Simplified 80.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.2 \cdot 10^{+82}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.9 \cdot 10^{+40}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -9.5 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.9 \cdot 10^{+40}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -9.5e+84)
     t_0
     (if (<= rand 1.9e+40) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -9.5e+84) {
		tmp = t_0;
	} else if (rand <= 1.9e+40) {
		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 <= (-9.5d+84)) then
        tmp = t_0
    else if (rand <= 1.9d+40) 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 <= -9.5e+84) {
		tmp = t_0;
	} else if (rand <= 1.9e+40) {
		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 <= -9.5e+84:
		tmp = t_0
	elif rand <= 1.9e+40:
		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 <= -9.5e+84)
		tmp = t_0;
	elseif (rand <= 1.9e+40)
		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 <= -9.5e+84)
		tmp = t_0;
	elseif (rand <= 1.9e+40)
		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, -9.5e+84], t$95$0, If[LessEqual[rand, 1.9e+40], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -9.49999999999999979e84 or 1.90000000000000002e40 < rand

   1. Initial program 99.5%

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

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

      10. distribute-lft-in N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 97.7

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

   7. Simplified 97.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 97.8

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

   9. Applied egg-rr 97.8%

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

   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 85.3

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

   12. Simplified 85.3%

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

   if -9.49999999999999979e84 < rand < 1.90000000000000002e40

   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 95.8

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

   5. Simplified 95.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -9.5 \cdot 10^{+84}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 1.9 \cdot 10^{+40}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. Simplified 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. lower-+.f64 N/A

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

   7. lower-fma.f64 99.8

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.8

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 7: 68.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 4.4 \cdot 10^{+156}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a + -0.3333333333333333\right) \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 4.4e+156)
   (+ a -0.3333333333333333)
   (/ (* (+ a -0.3333333333333333) rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 4.4e+156) {
		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 <= 4.4d+156) 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 <= 4.4e+156) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 4.4e+156:
		tmp = a + -0.3333333333333333
	else:
		tmp = ((a + -0.3333333333333333) * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 4.4e+156)
		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 <= 4.4e+156)
		tmp = a + -0.3333333333333333;
	else
		tmp = ((a + -0.3333333333333333) * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 4.4e+156], 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 < 4.40000000000000008e156

   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 68.7

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

   5. Simplified 68.7%

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

   if 4.40000000000000008e156 < 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 inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in rand around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 5.3

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

   8. Simplified 5.3%

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

      1. lift-+.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 37.3

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 37.3

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

   10. Applied egg-rr 37.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 4.4 \cdot 10^{+156}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a + -0.3333333333333333\right) \cdot rand}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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. +-commutative N/A

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

   10. distribute-lft-in N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 99.0

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

7. Simplified 99.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 99.1

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

Alternative 9: 98.7% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (a rand)
 :precision binary64
 (fma 0.3333333333333333 (fma (sqrt a) rand -1.0) a))

C

double code(double a, double rand) {
	return fma(0.3333333333333333, fma(sqrt(a), rand, -1.0), a);
}

Julia

function code(a, rand)
	return fma(0.3333333333333333, fma(sqrt(a), rand, -1.0), a)
end

Wolfram

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

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

   10. distribute-lft-in N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 99.0

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

7. Simplified 99.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 99.1

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in rand around 0

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

    1. associate--l+ N/A

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

    2. +-commutative N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. metadata-eval N/A

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

    8. rgt-mult-inverse N/A

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

    9. associate-*r* N/A

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

    10. metadata-eval N/A

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

    11. distribute-rgt-neg-in N/A

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

    12. *-commutative N/A

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

    13. distribute-lft-in N/A

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

    14. sub-neg N/A

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

    15. *-commutative N/A

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

    16. distribute-lft-out-- N/A

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

    17. associate-*l* N/A

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

    18. lower-fma.f64 N/A

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

12. Simplified 98.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(\sqrt{a}, rand, -1\right), a\right)} \]
13. Add Preprocessing

Alternative 10: 97.7% 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. +-commutative N/A

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

   10. distribute-lft-in N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 97.5

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

7. Simplified 97.5%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. lift-/.f64 N/A

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

   12. inv-pow N/A

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

   13. pow-pow N/A

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

   14. pow-plus N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. pow1/2 N/A

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

   18. lift-sqrt.f64 N/A

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

9. Applied egg-rr 97.6%

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

Alternative 11: 97.7% 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

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

   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. +-commutative N/A

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

   10. distribute-lft-in N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 97.5

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

7. Simplified 97.5%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

9. Applied egg-rr 97.5%

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

Alternative 12: 62.9% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 61.8

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

5. Simplified 61.8%

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

Alternative 13: 1.5% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 61.8

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

5. Simplified 61.8%

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

6. Taylor expanded in a around 0

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

8. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024207 
(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))))