Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.8%

Time: 12.3s

Alternatives: 15

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. sub-neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l/ N/A

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

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

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

   9. *-lft-identity N/A

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

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

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

   11. sub-neg N/A

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

   12. distribute-lft-in N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. accelerator-lowering-fma.f64 N/A

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

   18. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333}\\ \mathbf{if}\;rand \leq -2 \cdot 10^{+95}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot t\_0\right)\\ \mathbf{elif}\;rand \leq 1.55 \cdot 10^{+66}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (sqrt (+ a -0.3333333333333333))))
   (if (<= rand -2e+95)
     (* 0.3333333333333333 (* rand t_0))
     (if (<= rand 1.55e+66)
       (+ a -0.3333333333333333)
       (* t_0 (* rand 0.3333333333333333))))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333));
	double tmp;
	if (rand <= -2e+95) {
		tmp = 0.3333333333333333 * (rand * t_0);
	} else if (rand <= 1.55e+66) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (rand * 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a + (-0.3333333333333333d0)))
    if (rand <= (-2d+95)) then
        tmp = 0.3333333333333333d0 * (rand * t_0)
    else if (rand <= 1.55d+66) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * (rand * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a + -0.3333333333333333));
	double tmp;
	if (rand <= -2e+95) {
		tmp = 0.3333333333333333 * (rand * t_0);
	} else if (rand <= 1.55e+66) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (rand * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333))
	tmp = 0
	if rand <= -2e+95:
		tmp = 0.3333333333333333 * (rand * t_0)
	elif rand <= 1.55e+66:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (rand * 0.3333333333333333)
	return tmp

Julia

function code(a, rand)
	t_0 = sqrt(Float64(a + -0.3333333333333333))
	tmp = 0.0
	if (rand <= -2e+95)
		tmp = Float64(0.3333333333333333 * Float64(rand * t_0));
	elseif (rand <= 1.55e+66)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(rand * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a + -0.3333333333333333));
	tmp = 0.0;
	if (rand <= -2e+95)
		tmp = 0.3333333333333333 * (rand * t_0);
	elseif (rand <= 1.55e+66)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (rand * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[rand, -2e+95], N[(0.3333333333333333 * N[(rand * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 1.55e+66], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.00000000000000004e95

   1. Initial program 99.7%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 95.1

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

   5. Simplified 95.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. +-lowering-+.f64 95.3

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

   7. Applied egg-rr 95.3%

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

   if -2.00000000000000004e95 < rand < 1.55000000000000009e66

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

   if 1.55000000000000009e66 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2 \cdot 10^{+95}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a + -0.3333333333333333}\right)\\ \mathbf{elif}\;rand \leq 1.55 \cdot 10^{+66}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333}\\ \mathbf{if}\;rand \leq -1.95 \cdot 10^{+95}:\\ \;\;\;\;rand \cdot \left(t\_0 \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 7.8 \cdot 10^{+65}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (sqrt (+ a -0.3333333333333333))))
   (if (<= rand -1.95e+95)
     (* rand (* t_0 0.3333333333333333))
     (if (<= rand 7.8e+65)
       (+ a -0.3333333333333333)
       (* t_0 (* rand 0.3333333333333333))))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333));
	double tmp;
	if (rand <= -1.95e+95) {
		tmp = rand * (t_0 * 0.3333333333333333);
	} else if (rand <= 7.8e+65) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (rand * 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a + (-0.3333333333333333d0)))
    if (rand <= (-1.95d+95)) then
        tmp = rand * (t_0 * 0.3333333333333333d0)
    else if (rand <= 7.8d+65) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0 * (rand * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = Math.sqrt((a + -0.3333333333333333));
	double tmp;
	if (rand <= -1.95e+95) {
		tmp = rand * (t_0 * 0.3333333333333333);
	} else if (rand <= 7.8e+65) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0 * (rand * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333))
	tmp = 0
	if rand <= -1.95e+95:
		tmp = rand * (t_0 * 0.3333333333333333)
	elif rand <= 7.8e+65:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0 * (rand * 0.3333333333333333)
	return tmp

Julia

function code(a, rand)
	t_0 = sqrt(Float64(a + -0.3333333333333333))
	tmp = 0.0
	if (rand <= -1.95e+95)
		tmp = Float64(rand * Float64(t_0 * 0.3333333333333333));
	elseif (rand <= 7.8e+65)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(t_0 * Float64(rand * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a + -0.3333333333333333));
	tmp = 0.0;
	if (rand <= -1.95e+95)
		tmp = rand * (t_0 * 0.3333333333333333);
	elseif (rand <= 7.8e+65)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0 * (rand * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[rand, -1.95e+95], N[(rand * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 7.8e+65], N[(a + -0.3333333333333333), $MachinePrecision], N[(t$95$0 * N[(rand * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.9499999999999999e95

   1. Initial program 99.7%

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

   3. Taylor expanded in rand around inf

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

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

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. /-lowering-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in rand around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 95.2

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

   8. Simplified 95.2%

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

   if -1.9499999999999999e95 < rand < 7.7999999999999996e65

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

   if 7.7999999999999996e65 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.95 \cdot 10^{+95}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 7.8 \cdot 10^{+65}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a + -0.3333333333333333} \cdot \left(rand \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 2.5 \cdot 10^{+64}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* (sqrt (+ a -0.3333333333333333)) 0.3333333333333333))))
   (if (<= rand -2.8e+94)
     t_0
     (if (<= rand 2.5e+64) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	double tmp;
	if (rand <= -2.8e+94) {
		tmp = t_0;
	} else if (rand <= 2.5e+64) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = rand * (sqrt((a + (-0.3333333333333333d0))) * 0.3333333333333333d0)
    if (rand <= (-2.8d+94)) then
        tmp = t_0
    else if (rand <= 2.5d+64) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand * (Math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	double tmp;
	if (rand <= -2.8e+94) {
		tmp = t_0;
	} else if (rand <= 2.5e+64) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (math.sqrt((a + -0.3333333333333333)) * 0.3333333333333333)
	tmp = 0
	if rand <= -2.8e+94:
		tmp = t_0
	elif rand <= 2.5e+64:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(sqrt(Float64(a + -0.3333333333333333)) * 0.3333333333333333))
	tmp = 0.0
	if (rand <= -2.8e+94)
		tmp = t_0;
	elseif (rand <= 2.5e+64)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (sqrt((a + -0.3333333333333333)) * 0.3333333333333333);
	tmp = 0.0;
	if (rand <= -2.8e+94)
		tmp = t_0;
	elseif (rand <= 2.5e+64)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -2.8e+94], t$95$0, If[LessEqual[rand, 2.5e+64], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.79999999999999998e94 or 2.5e64 < rand

   1. Initial program 99.6%

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

   3. Taylor expanded in rand around inf

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

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

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. /-lowering-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in rand around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 97.1

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

   8. Simplified 97.1%

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

   if -2.79999999999999998e94 < rand < 2.5e64

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;rand \leq 2.5 \cdot 10^{+64}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(\sqrt{a + -0.3333333333333333} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -3.1 \cdot 10^{+94}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 2.5 \cdot 10^{+66}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -3.1e+94)
   (* rand (* 0.3333333333333333 (sqrt a)))
   (if (<= rand 2.5e+66)
     (+ a -0.3333333333333333)
     (* (* rand 0.3333333333333333) (sqrt a)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -3.1e+94) {
		tmp = rand * (0.3333333333333333 * sqrt(a));
	} else if (rand <= 2.5e+66) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-3.1d+94)) then
        tmp = rand * (0.3333333333333333d0 * sqrt(a))
    else if (rand <= 2.5d+66) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (rand * 0.3333333333333333d0) * sqrt(a)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -3.1e+94) {
		tmp = rand * (0.3333333333333333 * Math.sqrt(a));
	} else if (rand <= 2.5e+66) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (rand * 0.3333333333333333) * Math.sqrt(a);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -3.1e+94:
		tmp = rand * (0.3333333333333333 * math.sqrt(a))
	elif rand <= 2.5e+66:
		tmp = a + -0.3333333333333333
	else:
		tmp = (rand * 0.3333333333333333) * math.sqrt(a)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -3.1e+94)
		tmp = Float64(rand * Float64(0.3333333333333333 * sqrt(a)));
	elseif (rand <= 2.5e+66)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(rand * 0.3333333333333333) * sqrt(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -3.1e+94)
		tmp = rand * (0.3333333333333333 * sqrt(a));
	elseif (rand <= 2.5e+66)
		tmp = a + -0.3333333333333333;
	else
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -3.1e+94], N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 2.5e+66], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -3.09999999999999991e94

   1. Initial program 99.7%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 95.1

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

   5. Simplified 95.1%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 93.8

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

   8. Simplified 93.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 93.8

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

   10. Applied egg-rr 93.8%

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

   if -3.09999999999999991e94 < rand < 2.49999999999999996e66

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

   if 2.49999999999999996e66 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-lowering-sqrt.f64 97.1

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

   8. Simplified 97.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -3.1 \cdot 10^{+94}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 2.5 \cdot 10^{+66}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -4.4 \cdot 10^{+94}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+64}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -4.4e+94)
   (* 0.3333333333333333 (* rand (sqrt a)))
   (if (<= rand 2.4e+64)
     (+ a -0.3333333333333333)
     (* (* rand 0.3333333333333333) (sqrt a)))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -4.4e+94) {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	} else if (rand <= 2.4e+64) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-4.4d+94)) then
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    else if (rand <= 2.4d+64) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (rand * 0.3333333333333333d0) * sqrt(a)
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -4.4e+94) {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	} else if (rand <= 2.4e+64) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (rand * 0.3333333333333333) * Math.sqrt(a);
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -4.4e+94:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	elif rand <= 2.4e+64:
		tmp = a + -0.3333333333333333
	else:
		tmp = (rand * 0.3333333333333333) * math.sqrt(a)
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -4.4e+94)
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	elseif (rand <= 2.4e+64)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(rand * 0.3333333333333333) * sqrt(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -4.4e+94)
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	elseif (rand <= 2.4e+64)
		tmp = a + -0.3333333333333333;
	else
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -4.4e+94], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 2.4e+64], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -4.40000000000000024e94

   1. Initial program 99.7%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 95.1

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

   5. Simplified 95.1%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 93.8

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

   8. Simplified 93.8%

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

   if -4.40000000000000024e94 < rand < 2.39999999999999999e64

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

   if 2.39999999999999999e64 < rand

   1. Initial program 99.5%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-lowering-sqrt.f64 97.1

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

   8. Simplified 97.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -4.4 \cdot 10^{+94}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 2.4 \cdot 10^{+64}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -6.5 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 9.2 \cdot 10^{+63}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -6.5e+94)
     t_0
     (if (<= rand 9.2e+63) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -6.5e+94) {
		tmp = t_0;
	} else if (rand <= 9.2e+63) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (rand * sqrt(a))
    if (rand <= (-6.5d+94)) then
        tmp = t_0
    else if (rand <= 9.2d+63) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * Math.sqrt(a));
	double tmp;
	if (rand <= -6.5e+94) {
		tmp = t_0;
	} else if (rand <= 9.2e+63) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = 0.3333333333333333 * (rand * math.sqrt(a))
	tmp = 0
	if rand <= -6.5e+94:
		tmp = t_0
	elif rand <= 9.2e+63:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(0.3333333333333333 * Float64(rand * sqrt(a)))
	tmp = 0.0
	if (rand <= -6.5e+94)
		tmp = t_0;
	elseif (rand <= 9.2e+63)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = 0.3333333333333333 * (rand * sqrt(a));
	tmp = 0.0;
	if (rand <= -6.5e+94)
		tmp = t_0;
	elseif (rand <= 9.2e+63)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -6.5e+94], t$95$0, If[LessEqual[rand, 9.2e+63], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -6.49999999999999976e94 or 9.19999999999999973e63 < rand

   1. Initial program 99.6%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 95.4

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

   8. Simplified 95.4%

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

   if -6.49999999999999976e94 < rand < 9.19999999999999973e63

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 95.1

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

   5. Simplified 95.1%

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

Alternative 8: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-lowering-+.f64 99.8

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

5. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.8

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 9: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-lowering-+.f64 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in a around inf

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 10: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-lowering-+.f64 99.8

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

5. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.8

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in a around inf

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

   1. sqrt-lowering-sqrt.f64 99.0

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 11: 97.9% accurate, 2.8× 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. Add Preprocessing

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-lowering-+.f64 99.8

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

5. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.8

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in a around inf

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

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

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

   2. *-commutative N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 98.3

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 12: 68.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 1.1 \cdot 10^{+150}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 1.1e+150) (+ a -0.3333333333333333) (/ (* a rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 1.1e+150) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= 1.1d+150) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = (a * rand) / rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= 1.1e+150) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 1.1e+150:
		tmp = a + -0.3333333333333333
	else:
		tmp = (a * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 1.1e+150)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(Float64(a * rand) / rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= 1.1e+150)
		tmp = a + -0.3333333333333333;
	else
		tmp = (a * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 1.1e+150], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(a * rand), $MachinePrecision] / rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 1.1e150

   1. Initial program 99.9%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 71.5

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

   5. Simplified 71.5%

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

   if 1.1e150 < rand

   1. Initial program 99.6%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-lowering-+.f64 6.2

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

   5. Simplified 6.2%

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. +-lowering-+.f64 49.3

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

   7. Applied egg-rr 49.3%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 49.3

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

   10. Simplified 49.3%

       \[\leadsto \frac{\color{blue}{rand \cdot a}}{rand} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 1.1 \cdot 10^{+150}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.7% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-lowering-+.f64 62.5

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

5. Simplified 62.5%

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

Alternative 14: 61.7% accurate, 68.0× 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. Add Preprocessing

3. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-lowering-+.f64 62.5

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

5. Simplified 62.5%

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

6. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a} \]
7. Step-by-step derivation

8. Simplified 61.9%

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

Alternative 15: 1.6% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-lowering-+.f64 62.5

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

5. Simplified 62.5%

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

6. Taylor expanded in a around 0

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

8. Simplified 1.5%

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

Reproduce

herbie shell --seed 2024203 
(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))))