Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 10.8s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot \sqrt{a - \frac{1}{3}}\right) \cdot rand\right)\right), \color{blue}{\left(a - \frac{1}{3}\right)}\right) \]

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-0.3333333333333333 + a} \cdot \frac{rand}{3}\\ \mathbf{if}\;rand \leq -7.8 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2.2 \cdot 10^{+76}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ -0.3333333333333333 a)) (/ rand 3.0))))
   (if (<= rand -7.8e+89)
     t_0
     (if (<= rand 2.2e+76) (+ -0.3333333333333333 a) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((-0.3333333333333333 + a)) * (rand / 3.0);
	double tmp;
	if (rand <= -7.8e+89) {
		tmp = t_0;
	} else if (rand <= 2.2e+76) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((-0.3333333333333333d0) + a)) * (rand / 3.0d0)
    if (rand <= (-7.8d+89)) then
        tmp = t_0
    else if (rand <= 2.2d+76) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((-0.3333333333333333 + a)) * (rand / 3.0);
	double tmp;
	if (rand <= -7.8e+89) {
		tmp = t_0;
	} else if (rand <= 2.2e+76) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((-0.3333333333333333 + a)) * (rand / 3.0)
	tmp = 0
	if rand <= -7.8e+89:
		tmp = t_0
	elif rand <= 2.2e+76:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(-0.3333333333333333 + a)) * Float64(rand / 3.0))
	tmp = 0.0
	if (rand <= -7.8e+89)
		tmp = t_0;
	elseif (rand <= 2.2e+76)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((-0.3333333333333333 + a)) * (rand / 3.0);
	tmp = 0.0;
	if (rand <= -7.8e+89)
		tmp = t_0;
	elseif (rand <= 2.2e+76)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[N[(-0.3333333333333333 + a), $MachinePrecision]], $MachinePrecision] * N[(rand / 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -7.8e+89], t$95$0, If[LessEqual[rand, 2.2e+76], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -7.80000000000000021e89 or 2.2e76 < 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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{a - \frac{1}{3}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a - \frac{1}{3}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \frac{-1}{3}\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{3} + a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      9. *-lowering-*.f64 91.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right) \]

   7. Simplified 91.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(rand \cdot \color{blue}{\frac{1}{3}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(rand \cdot \frac{1}{\color{blue}{3}}\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{rand}{\color{blue}{3}}\right)\right) \]

      4. /-lowering-/.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(rand, \color{blue}{3}\right)\right) \]

   9. Applied egg-rr 91.1%

      \[\leadsto \sqrt{-0.3333333333333333 + a} \cdot \color{blue}{\frac{rand}{3}} \]

   if -7.80000000000000021e89 < rand < 2.2e76

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 93.5%

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

Alternative 3: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-0.3333333333333333 + a} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{if}\;rand \leq -8.8 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2.15 \cdot 10^{+76}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ -0.3333333333333333 a)) (* 0.3333333333333333 rand))))
   (if (<= rand -8.8e+89)
     t_0
     (if (<= rand 2.15e+76) (+ -0.3333333333333333 a) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((-0.3333333333333333 + a)) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -8.8e+89) {
		tmp = t_0;
	} else if (rand <= 2.15e+76) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((-0.3333333333333333d0) + a)) * (0.3333333333333333d0 * rand)
    if (rand <= (-8.8d+89)) then
        tmp = t_0
    else if (rand <= 2.15d+76) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((-0.3333333333333333 + a)) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -8.8e+89) {
		tmp = t_0;
	} else if (rand <= 2.15e+76) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((-0.3333333333333333 + a)) * (0.3333333333333333 * rand)
	tmp = 0
	if rand <= -8.8e+89:
		tmp = t_0
	elif rand <= 2.15e+76:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(-0.3333333333333333 + a)) * Float64(0.3333333333333333 * rand))
	tmp = 0.0
	if (rand <= -8.8e+89)
		tmp = t_0;
	elseif (rand <= 2.15e+76)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((-0.3333333333333333 + a)) * (0.3333333333333333 * rand);
	tmp = 0.0;
	if (rand <= -8.8e+89)
		tmp = t_0;
	elseif (rand <= 2.15e+76)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[N[(-0.3333333333333333 + a), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 * rand), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.8e+89], t$95$0, If[LessEqual[rand, 2.15e+76], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -8.8000000000000001e89 or 2.14999999999999989e76 < 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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{a - \frac{1}{3}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a - \frac{1}{3}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \frac{-1}{3}\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{3} + a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      9. *-lowering-*.f64 91.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right) \]

   7. Simplified 91.0%

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

   if -8.8000000000000001e89 < rand < 2.14999999999999989e76

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 93.5%

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

Alternative 4: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -2.3 \cdot 10^{+94}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 4.3 \cdot 10^{+75}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a} \cdot \frac{rand}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -2.3e+94)
   (* 0.3333333333333333 (* rand (sqrt a)))
   (if (<= rand 4.3e+75) (+ -0.3333333333333333 a) (* (sqrt a) (/ rand 3.0)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -2.3e+94) {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	} else if (rand <= 4.3e+75) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = sqrt(a) * (rand / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-2.3d+94)) then
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    else if (rand <= 4.3d+75) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = sqrt(a) * (rand / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -2.3e+94) {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	} else if (rand <= 4.3e+75) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = Math.sqrt(a) * (rand / 3.0);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -2.3e+94:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	elif rand <= 4.3e+75:
		tmp = -0.3333333333333333 + a
	else:
		tmp = math.sqrt(a) * (rand / 3.0)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -2.3e+94)
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	elseif (rand <= 4.3e+75)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(sqrt(a) * Float64(rand / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -2.3e+94)
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	elseif (rand <= 4.3e+75)
		tmp = -0.3333333333333333 + a;
	else
		tmp = sqrt(a) * (rand / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -2.3e+94], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.3e+75], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[Sqrt[a], $MachinePrecision] * N[(rand / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.3e94

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{a - \frac{1}{3}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a - \frac{1}{3}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \frac{-1}{3}\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{3} + a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      9. *-lowering-*.f64 88.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right) \]

   7. Simplified 88.5%

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

   8. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{a} \cdot rand\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(rand \cdot \color{blue}{\sqrt{a}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \color{blue}{\left(\sqrt{a}\right)}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 85.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \mathsf{sqrt.f64}\left(a\right)\right)\right) \]

   10. Simplified 85.2%

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

   if -2.3e94 < rand < 4.3000000000000001e75

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 92.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 92.9%

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

   if 4.3000000000000001e75 < 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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.4%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{a - \frac{1}{3}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a - \frac{1}{3}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \frac{-1}{3}\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{3} + a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      9. *-lowering-*.f64 95.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right) \]

   7. Simplified 95.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(rand \cdot \color{blue}{\frac{1}{3}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(rand \cdot \frac{1}{\color{blue}{3}}\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{rand}{\color{blue}{3}}\right)\right) \]

      4. /-lowering-/.f64 95.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{/.f64}\left(rand, \color{blue}{3}\right)\right) \]

   9. Applied egg-rr 95.9%

      \[\leadsto \sqrt{-0.3333333333333333 + a} \cdot \color{blue}{\frac{rand}{3}} \]

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{a}\right)}, \mathsf{/.f64}\left(rand, 3\right)\right) \]
   11. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 94.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(a\right), \mathsf{/.f64}\left(\color{blue}{rand}, 3\right)\right) \]

   12. Simplified 94.1%

       \[\leadsto \color{blue}{\sqrt{a}} \cdot \frac{rand}{3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -2.1 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 3.2 \cdot 10^{+75}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -2.1e+94)
     t_0
     (if (<= rand 3.2e+75) (+ -0.3333333333333333 a) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -2.1e+94) {
		tmp = t_0;
	} else if (rand <= 3.2e+75) {
		tmp = -0.3333333333333333 + a;
	} 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 <= (-2.1d+94)) then
        tmp = t_0
    else if (rand <= 3.2d+75) then
        tmp = (-0.3333333333333333d0) + a
    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 <= -2.1e+94) {
		tmp = t_0;
	} else if (rand <= 3.2e+75) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.3333333333333333 * (rand * math.sqrt(a))
	tmp = 0
	if rand <= -2.1e+94:
		tmp = t_0
	elif rand <= 3.2e+75:
		tmp = -0.3333333333333333 + a
	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 <= -2.1e+94)
		tmp = t_0;
	elseif (rand <= 3.2e+75)
		tmp = Float64(-0.3333333333333333 + a);
	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 <= -2.1e+94)
		tmp = t_0;
	elseif (rand <= 3.2e+75)
		tmp = -0.3333333333333333 + a;
	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, -2.1e+94], t$95$0, If[LessEqual[rand, 3.2e+75], N[(-0.3333333333333333 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.09999999999999989e94 or 3.19999999999999985e75 < 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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{a - \frac{1}{3}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a - \frac{1}{3}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(a + \frac{-1}{3}\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{-1}{3} + a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right) \]

      9. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right) \]

   7. Simplified 91.8%

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

   8. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{a} \cdot rand\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(rand \cdot \color{blue}{\sqrt{a}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \color{blue}{\left(\sqrt{a}\right)}\right)\right) \]

      4. sqrt-lowering-sqrt.f64 89.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \mathsf{sqrt.f64}\left(a\right)\right)\right) \]

   10. Simplified 89.2%

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

   if -2.09999999999999989e94 < rand < 3.19999999999999985e75

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 92.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 92.9%

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

Alternative 6: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = ((-0.3333333333333333d0) + a) * (1.0d0 + (rand / (3.0d0 * sqrt(a))))
end function

Java

public static double code(double a, double rand) {
	return (-0.3333333333333333 + a) * (1.0 + (rand / (3.0 * Math.sqrt(a))));
}

Python

def code(a, rand):
	return (-0.3333333333333333 + a) * (1.0 + (rand / (3.0 * math.sqrt(a))))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (-0.3333333333333333 + a) * (1.0 + (rand / (3.0 * sqrt(a))));
end

Wolfram

code[a_, rand_] := N[(N[(-0.3333333333333333 + a), $MachinePrecision] * N[(1.0 + N[(rand / N[(3.0 * N[Sqrt[a], $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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in a around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(3 \cdot \sqrt{a}\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{*.f64}\left(3, \color{blue}{\left(\sqrt{a}\right)}\right)\right)\right)\right) \]

   2. sqrt-lowering-sqrt.f64 98.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(a\right)\right)\right)\right)\right) \]

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 7: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\left(\frac{-1}{3} \cdot \sqrt{a - \frac{1}{3}}\right) \cdot rand\right)\right), \color{blue}{\left(a - \frac{1}{3}\right)}\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in a around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\sqrt{a}\right)}, \mathsf{*.f64}\left(\frac{1}{3}, rand\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right) \]
9. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 98.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(a\right), \mathsf{*.f64}\left(\frac{1}{3}, rand\right)\right), \mathsf{+.f64}\left(\frac{-1}{3}, a\right)\right) \]

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 8: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return a + (0.3333333333333333 * (rand * math.sqrt(a)))

Julia

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

MATLAB

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

Wolfram

code[a_, rand_] := N[(a + N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

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

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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(1 + \frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{\frac{1}{a}} \cdot rand\right)\right)}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{3} \cdot \sqrt{\frac{1}{a}}\right) \cdot \color{blue}{rand}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{1}{3}\right) \cdot rand\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{1}{a}} \cdot \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{a}}\right), \color{blue}{\left(\frac{1}{3} \cdot rand\right)}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{a}\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot rand\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \left(\frac{1}{3} \cdot rand\right)\right)\right)\right) \]

   9. *-lowering-*.f64 97.7%

      \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, a\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{rand}\right)\right)\right)\right) \]

7. Simplified 97.7%

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

8. Taylor expanded in a around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(a, \color{blue}{\left(\frac{1}{3} \cdot \left(\sqrt{a} \cdot rand\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{a} \cdot rand\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \left(rand \cdot \color{blue}{\sqrt{a}}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \color{blue}{\left(\sqrt{a}\right)}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 97.8%

      \[\leadsto \mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(rand, \mathsf{sqrt.f64}\left(a\right)\right)\right)\right) \]

10. Simplified 97.8%

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

Alternative 9: 74.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.1111111111111111 - a \cdot a\\ \mathbf{if}\;rand \leq -3.6 \cdot 10^{+137}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 6.5 \cdot 10^{+152}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- 0.1111111111111111 (* a a))))
   (if (<= rand -3.6e+137)
     (* t_0 (+ (* a 9.0) -3.0))
     (if (<= rand 6.5e+152)
       (+ -0.3333333333333333 a)
       (* t_0 (+ -3.0 (* a (+ 9.0 (* a -27.0)))))))))

C

double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -3.6e+137) {
		tmp = t_0 * ((a * 9.0) + -3.0);
	} else if (rand <= 6.5e+152) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	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 = 0.1111111111111111d0 - (a * a)
    if (rand <= (-3.6d+137)) then
        tmp = t_0 * ((a * 9.0d0) + (-3.0d0))
    else if (rand <= 6.5d+152) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0 * ((-3.0d0) + (a * (9.0d0 + (a * (-27.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -3.6e+137) {
		tmp = t_0 * ((a * 9.0) + -3.0);
	} else if (rand <= 6.5e+152) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.1111111111111111 - (a * a)
	tmp = 0
	if rand <= -3.6e+137:
		tmp = t_0 * ((a * 9.0) + -3.0)
	elif rand <= 6.5e+152:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))))
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.1111111111111111 - Float64(a * a))
	tmp = 0.0
	if (rand <= -3.6e+137)
		tmp = Float64(t_0 * Float64(Float64(a * 9.0) + -3.0));
	elseif (rand <= 6.5e+152)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(t_0 * Float64(-3.0 + Float64(a * Float64(9.0 + Float64(a * -27.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.1111111111111111 - (a * a);
	tmp = 0.0;
	if (rand <= -3.6e+137)
		tmp = t_0 * ((a * 9.0) + -3.0);
	elseif (rand <= 6.5e+152)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0 * (-3.0 + (a * (9.0 + (a * -27.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -3.6e+137], N[(t$95$0 * N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 6.5e+152], N[(-0.3333333333333333 + a), $MachinePrecision], N[(t$95$0 * N[(-3.0 + N[(a * N[(9.0 + N[(a * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -3.6e137

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(9 \cdot a - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(9 \cdot a + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(9 \cdot a\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot 9\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, 9\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       5. metadata-eval 53.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, 9\right), -3\right)\right) \]

   12. Simplified 53.9%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot 9 + -3\right)} \]

   if -3.6e137 < rand < 6.4999999999999997e152

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 78.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 78.9%

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

   if 6.4999999999999997e152 < rand

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.5%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 31.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(a \cdot \left(9 + -27 \cdot a\right) - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(a \cdot \left(9 + -27 \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(9 + -27 \cdot a\right)\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(9 + -27 \cdot a\right)\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(-27 \cdot a\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \left(a \cdot -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), \left(\mathsf{neg}\left(3\right)\right)\right)\right) \]

       7. metadata-eval 48.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(9, \mathsf{*.f64}\left(a, -27\right)\right)\right), -3\right)\right) \]

   12. Simplified 48.3%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(9 + a \cdot -27\right) + -3\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -3.6 \cdot 10^{+137}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 6.5 \cdot 10^{+152}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot \left(-3 + a \cdot \left(9 + a \cdot -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.1111111111111111 - a \cdot a\\ \mathbf{if}\;rand \leq -1.8 \cdot 10^{+140}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot 9 + -3\right)\\ \mathbf{elif}\;rand \leq 1.36 \cdot 10^{+147}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (- 0.1111111111111111 (* a a))))
   (if (<= rand -1.8e+140)
     (* t_0 (+ (* a 9.0) -3.0))
     (if (<= rand 1.36e+147) (+ -0.3333333333333333 a) (* t_0 -3.0)))))

C

double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -1.8e+140) {
		tmp = t_0 * ((a * 9.0) + -3.0);
	} else if (rand <= 1.36e+147) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * -3.0;
	}
	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 = 0.1111111111111111d0 - (a * a)
    if (rand <= (-1.8d+140)) then
        tmp = t_0 * ((a * 9.0d0) + (-3.0d0))
    else if (rand <= 1.36d+147) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = t_0 * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.1111111111111111 - (a * a);
	double tmp;
	if (rand <= -1.8e+140) {
		tmp = t_0 * ((a * 9.0) + -3.0);
	} else if (rand <= 1.36e+147) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = t_0 * -3.0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.1111111111111111 - (a * a)
	tmp = 0
	if rand <= -1.8e+140:
		tmp = t_0 * ((a * 9.0) + -3.0)
	elif rand <= 1.36e+147:
		tmp = -0.3333333333333333 + a
	else:
		tmp = t_0 * -3.0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.1111111111111111 - Float64(a * a))
	tmp = 0.0
	if (rand <= -1.8e+140)
		tmp = Float64(t_0 * Float64(Float64(a * 9.0) + -3.0));
	elseif (rand <= 1.36e+147)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(t_0 * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.1111111111111111 - (a * a);
	tmp = 0.0;
	if (rand <= -1.8e+140)
		tmp = t_0 * ((a * 9.0) + -3.0);
	elseif (rand <= 1.36e+147)
		tmp = -0.3333333333333333 + a;
	else
		tmp = t_0 * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.8e+140], N[(t$95$0 * N[(N[(a * 9.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.36e+147], N[(-0.3333333333333333 + a), $MachinePrecision], N[(t$95$0 * -3.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.8e140

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 0.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 0.4%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 0.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{\left(9 \cdot a - 3\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(9 \cdot a + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(9 \cdot a\right), \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\left(a \cdot 9\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, 9\right), \left(\mathsf{neg}\left(\color{blue}{3}\right)\right)\right)\right) \]

       5. metadata-eval 53.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, 9\right), -3\right)\right) \]

   12. Simplified 53.9%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{\left(a \cdot 9 + -3\right)} \]

   if -1.8e140 < rand < 1.35999999999999996e147

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 79.7%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 79.7%

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

   if 1.35999999999999996e147 < rand

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{-3}\right) \]
   11. Step-by-step derivation

   12. Simplified 30.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{-3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.7% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 4.4 \cdot 10^{+148}:\\ \;\;\;\;-0.3333333333333333 + a\\ \mathbf{else}:\\ \;\;\;\;\left(0.1111111111111111 - a \cdot a\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 4.4e+148)
   (+ -0.3333333333333333 a)
   (* (- 0.1111111111111111 (* a a)) -3.0)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 4.4e+148) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	}
	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+148) then
        tmp = (-0.3333333333333333d0) + a
    else
        tmp = (0.1111111111111111d0 - (a * a)) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 4.4e+148) {
		tmp = -0.3333333333333333 + a;
	} else {
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 4.4e+148:
		tmp = -0.3333333333333333 + a
	else:
		tmp = (0.1111111111111111 - (a * a)) * -3.0
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 4.4e+148)
		tmp = Float64(-0.3333333333333333 + a);
	else
		tmp = Float64(Float64(0.1111111111111111 - Float64(a * a)) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 4.4e+148)
		tmp = -0.3333333333333333 + a;
	else
		tmp = (0.1111111111111111 - (a * a)) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 4.4e+148], N[(-0.3333333333333333 + a), $MachinePrecision], N[(N[(0.1111111111111111 - N[(a * a), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 4.3999999999999998e148

   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. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 65.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 65.6%

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

   if 4.3999999999999998e148 < rand

   1. Initial program 99.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

   7. Simplified 5.6%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - a \cdot a\right), \color{blue}{\left(\frac{1}{\frac{-1}{3} - a}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{3} \cdot \frac{-1}{3}\right), \left(a \cdot a\right)\right), \left(\frac{\color{blue}{1}}{\frac{-1}{3} - a}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \left(a \cdot a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{\frac{-1}{3} - a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{3} - a\right)}\right)\right) \]

      8. --lowering--.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{-1}{3}, \color{blue}{a}\right)\right)\right) \]

   9. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - a \cdot a\right) \cdot \frac{1}{-0.3333333333333333 - a}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(a, a\right)\right), \color{blue}{-3}\right) \]
   11. Step-by-step derivation

   12. Simplified 30.5%

       \[\leadsto \left(0.1111111111111111 - a \cdot a\right) \cdot \color{blue}{-3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.5% accurate, 39.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, rand_] := N[(-0.3333333333333333 + 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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 58.4%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 58.4%

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

Alternative 13: 61.5% accurate, 119.0× speedup

Math

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

FPCore

(FPCore (a rand) :precision binary64 a)

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return a

Julia

function code(a, rand)
	return a
end

MATLAB

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

Wolfram

code[a_, rand_] := a

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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 58.4%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 58.4%

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

8. Taylor expanded in a around inf

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

10. Simplified 57.8%

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

Alternative 14: 1.5% accurate, 119.0× 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. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(\color{blue}{1} + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{1 \cdot rand}{\color{blue}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \left(\frac{rand}{\sqrt{\color{blue}{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \color{blue}{\left(\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(9 \cdot \left(a - \frac{1}{3}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\left(\left(a - \frac{1}{3}\right) \cdot 9\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a - \frac{1}{3}\right), 9\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right), 9\right)\right)\right)\right)\right) \]

   16. metadata-eval 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(rand, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \frac{-1}{3}\right), 9\right)\right)\right)\right)\right) \]

3. Simplified 99.8%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 58.4%

      \[\leadsto \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{a}\right) \]

7. Simplified 58.4%

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

8. Taylor expanded in a around 0

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

10. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024145 
(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))))