Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 9.3s

Alternatives: 9

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Alternative 2: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{a} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{elif}\;rand \leq 3.5 \cdot 10^{+108}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -8.5e+97)
   (* (sqrt a) (* 0.3333333333333333 rand))
   (if (<= rand 3.5e+108)
     (+ a -0.3333333333333333)
     (* rand (* 0.3333333333333333 (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -8.5e+97) {
		tmp = sqrt(a) * (0.3333333333333333 * rand);
	} else if (rand <= 3.5e+108) {
		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 <= (-8.5d+97)) then
        tmp = sqrt(a) * (0.3333333333333333d0 * rand)
    else if (rand <= 3.5d+108) 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 <= -8.5e+97) {
		tmp = Math.sqrt(a) * (0.3333333333333333 * rand);
	} else if (rand <= 3.5e+108) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = rand * (0.3333333333333333 * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -8.5e+97:
		tmp = math.sqrt(a) * (0.3333333333333333 * rand)
	elif rand <= 3.5e+108:
		tmp = a + -0.3333333333333333
	else:
		tmp = rand * (0.3333333333333333 * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -8.5e+97)
		tmp = Float64(sqrt(a) * Float64(0.3333333333333333 * rand));
	elseif (rand <= 3.5e+108)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = Float64(rand * Float64(0.3333333333333333 * sqrt(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -8.5e+97)
		tmp = sqrt(a) * (0.3333333333333333 * rand);
	elseif (rand <= 3.5e+108)
		tmp = a + -0.3333333333333333;
	else
		tmp = rand * (0.3333333333333333 * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -8.5e+97], N[(N[Sqrt[a], $MachinePrecision] * N[(0.3333333333333333 * rand), $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 3.5e+108], N[(a + -0.3333333333333333), $MachinePrecision], N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -8.4999999999999993e97

   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 a around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 94.1

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

   8. Simplified 94.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 94.1

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

   10. Applied egg-rr 94.1%

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

   if -8.4999999999999993e97 < rand < 3.5000000000000002e108

   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 92.9

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

   5. Simplified 92.9%

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

   if 3.5000000000000002e108 < rand

   1. Initial program 99.4%

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

   3. Taylor expanded in a around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 96.1

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

   5. Simplified 96.1%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 94.3

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

   8. Simplified 94.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{a} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{elif}\;rand \leq 3.5 \cdot 10^{+108}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -2.95 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 3.5 \cdot 10^{+108}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* 0.3333333333333333 (sqrt a)))))
   (if (<= rand -2.95e+97)
     t_0
     (if (<= rand 3.5e+108) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * sqrt(a));
	double tmp;
	if (rand <= -2.95e+97) {
		tmp = t_0;
	} else if (rand <= 3.5e+108) {
		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 <= (-2.95d+97)) then
        tmp = t_0
    else if (rand <= 3.5d+108) 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 <= -2.95e+97) {
		tmp = t_0;
	} else if (rand <= 3.5e+108) {
		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 <= -2.95e+97:
		tmp = t_0
	elif rand <= 3.5e+108:
		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 <= -2.95e+97)
		tmp = t_0;
	elseif (rand <= 3.5e+108)
		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 <= -2.95e+97)
		tmp = t_0;
	elseif (rand <= 3.5e+108)
		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, -2.95e+97], t$95$0, If[LessEqual[rand, 3.5e+108], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.95000000000000005e97 or 3.5000000000000002e108 < 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 a around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 97.6

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

   5. Simplified 97.6%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 94.2

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

   8. Simplified 94.2%

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

   if -2.95000000000000005e97 < rand < 3.5000000000000002e108

   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 92.9

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

   5. Simplified 92.9%

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

Alternative 4: 98.8% accurate, 2.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   7. +-lowering-+.f64 99.5

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in a around inf

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

10. Simplified 98.7%

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

Alternative 5: 97.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around inf

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 98.2

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

5. Simplified 98.2%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. pow1/2 N/A

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

   5. inv-pow N/A

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

   6. pow-pow N/A

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

   7. pow-plus N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. pow1/2 N/A

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

   11. *-lft-identity N/A

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

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

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

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

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

   14. sqrt-lowering-sqrt.f64 98.3

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

7. Applied egg-rr 98.3%

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

Alternative 6: 97.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 98.2

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

5. Simplified 98.2%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 98.2

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

8. Simplified 98.2%

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

Alternative 7: 62.6% 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 8: 61.6% 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 60.9%

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

Alternative 9: 1.5% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

3. 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 1.4%

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

Reproduce

herbie shell --seed 2024197 
(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))))