Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.7%

Time: 11.9s

Alternatives: 12

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-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 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 91.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -8.2 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* 0.3333333333333333 (sqrt a)))))
   (if (<= rand -8.2e+96)
     t_0
     (if (<= rand 3.5e+77) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * sqrt(a));
	double tmp;
	if (rand <= -8.2e+96) {
		tmp = t_0;
	} else if (rand <= 3.5e+77) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = rand * (0.3333333333333333d0 * sqrt(a))
    if (rand <= (-8.2d+96)) then
        tmp = t_0
    else if (rand <= 3.5d+77) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * Math.sqrt(a));
	double tmp;
	if (rand <= -8.2e+96) {
		tmp = t_0;
	} else if (rand <= 3.5e+77) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (0.3333333333333333 * math.sqrt(a))
	tmp = 0
	if rand <= -8.2e+96:
		tmp = t_0
	elif rand <= 3.5e+77:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(0.3333333333333333 * sqrt(a)))
	tmp = 0.0
	if (rand <= -8.2e+96)
		tmp = t_0;
	elseif (rand <= 3.5e+77)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (0.3333333333333333 * sqrt(a));
	tmp = 0.0;
	if (rand <= -8.2e+96)
		tmp = t_0;
	elseif (rand <= 3.5e+77)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.2e+96], t$95$0, If[LessEqual[rand, 3.5e+77], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -8.19999999999999996e96 or 3.5000000000000001e77 < 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 91.7

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

   5. Simplified 91.7%

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 91.8

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

   7. Applied egg-rr 91.8%

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

   8. Taylor expanded in a around inf

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

      1. sqrt-lowering-sqrt.f64 89.6

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

   10. Simplified 89.6%

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

   if -8.19999999999999996e96 < rand < 3.5000000000000001e77

   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 94.5

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

   5. Simplified 94.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8.2 \cdot 10^{+96}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{elif}\;rand \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -1.3 \cdot 10^{+101}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -1.3e+101)
   (* (* rand 0.3333333333333333) (sqrt a))
   (if (<= rand 4.5e+77)
     (+ a -0.3333333333333333)
     (* 0.3333333333333333 (* rand (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.3e+101) {
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	} else if (rand <= 4.5e+77) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-1.3d+101)) then
        tmp = (rand * 0.3333333333333333d0) * sqrt(a)
    else if (rand <= 4.5d+77) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = 0.3333333333333333d0 * (rand * sqrt(a))
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.3e+101) {
		tmp = (rand * 0.3333333333333333) * Math.sqrt(a);
	} else if (rand <= 4.5e+77) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 * (rand * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.3e+101:
		tmp = (rand * 0.3333333333333333) * math.sqrt(a)
	elif rand <= 4.5e+77:
		tmp = a + -0.3333333333333333
	else:
		tmp = 0.3333333333333333 * (rand * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.3e+101)
		tmp = Float64(Float64(rand * 0.3333333333333333) * sqrt(a));
	elseif (rand <= 4.5e+77)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 * Float64(rand * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.3e+101)
		tmp = (rand * 0.3333333333333333) * sqrt(a);
	elseif (rand <= 4.5e+77)
		tmp = a + -0.3333333333333333;
	else
		tmp = 0.3333333333333333 * (rand * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.3e+101], N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 4.5e+77], N[(a + -0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 * N[(rand * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.3e101

   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 93.0

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

   5. Simplified 93.0%

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

   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 90.7

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

   8. Simplified 90.7%

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

   if -1.3e101 < rand < 4.50000000000000024e77

   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 94.5

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

   5. Simplified 94.5%

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

   if 4.50000000000000024e77 < 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 90.4

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

   5. Simplified 90.4%

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

   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 88.3

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

   8. Simplified 88.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -1.3 \cdot 10^{+101}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -1 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.7 \cdot 10^{+77}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* rand (sqrt a)))))
   (if (<= rand -1e+97)
     t_0
     (if (<= rand 1.7e+77) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = 0.3333333333333333 * (rand * sqrt(a));
	double tmp;
	if (rand <= -1e+97) {
		tmp = t_0;
	} else if (rand <= 1.7e+77) {
		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 <= (-1d+97)) then
        tmp = t_0
    else if (rand <= 1.7d+77) 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 <= -1e+97) {
		tmp = t_0;
	} else if (rand <= 1.7e+77) {
		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 <= -1e+97:
		tmp = t_0
	elif rand <= 1.7e+77:
		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 <= -1e+97)
		tmp = t_0;
	elseif (rand <= 1.7e+77)
		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 <= -1e+97)
		tmp = t_0;
	elseif (rand <= 1.7e+77)
		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, -1e+97], t$95$0, If[LessEqual[rand, 1.7e+77], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.0000000000000001e97 or 1.69999999999999998e77 < 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 91.7

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

   5. Simplified 91.7%

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

   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 89.5

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

   8. Simplified 89.5%

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

   if -1.0000000000000001e97 < rand < 1.69999999999999998e77

   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 94.5

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

   5. Simplified 94.5%

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

Alternative 5: 99.8% accurate, 2.4× speedup

Math

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

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 6: 98.6% 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 99.3%

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

9. Final simplification 99.3%

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

Alternative 7: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l/ N/A

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

   5. sqrt-prod N/A

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

   6. metadata-eval N/A

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

   7. times-frac N/A

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

   8. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

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

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

   3. sqrt-lowering-sqrt.f64 98.6

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

7. Simplified 98.6%

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

Alternative 8: 68.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq 1.5 \cdot 10^{+145}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand 1.5e+145) (+ a -0.3333333333333333) (/ (* a rand) rand)))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= 1.5e+145) {
		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 <= 1.5d+145) 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 <= 1.5e+145) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = (a * rand) / rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= 1.5e+145:
		tmp = a + -0.3333333333333333
	else:
		tmp = (a * rand) / rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= 1.5e+145)
		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 <= 1.5e+145)
		tmp = a + -0.3333333333333333;
	else
		tmp = (a * rand) / rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, 1.5e+145], N[(a + -0.3333333333333333), $MachinePrecision], N[(N[(a * rand), $MachinePrecision] / rand), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < 1.5000000000000001e145

   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 70.4

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

   5. Simplified 70.4%

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

   if 1.5000000000000001e145 < 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 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 5.3

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

   5. Simplified 5.3%

      \[\leadsto \color{blue}{a + -0.3333333333333333} \]
   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 53.2

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

   7. Applied egg-rr 53.2%

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

   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 53.2

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

   10. Simplified 53.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq 1.5 \cdot 10^{+145}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot rand}{rand}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[Sqrt[a], $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 99.3%

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

9. Taylor expanded in a around inf

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

    1. sqrt-lowering-sqrt.f64 98.1

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

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

Alternative 10: 62.4% 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 61.8

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

5. Simplified 61.8%

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

Alternative 11: 61.2% 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 61.8

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

5. Simplified 61.8%

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

6. Taylor expanded in a around inf

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

8. Simplified 61.3%

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

Alternative 12: 1.5% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-lowering-+.f64 61.8

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

5. Simplified 61.8%

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

6. Taylor expanded in a around 0

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

8. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024198 
(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))))