Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 10.0s

Alternatives: 5

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 99.5%

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

3. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.8

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

5. Applied rewrites 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, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a + -0.3333333333333333} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \mathbf{if}\;rand \leq -1 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 10^{+112}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (sqrt (+ a -0.3333333333333333)) (* 0.3333333333333333 rand))))
   (if (<= rand -1e+91)
     t_0
     (if (<= rand 1e+112) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = sqrt((a + -0.3333333333333333)) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -1e+91) {
		tmp = t_0;
	} else if (rand <= 1e+112) {
		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 = sqrt((a + (-0.3333333333333333d0))) * (0.3333333333333333d0 * rand)
    if (rand <= (-1d+91)) then
        tmp = t_0
    else if (rand <= 1d+112) 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 = Math.sqrt((a + -0.3333333333333333)) * (0.3333333333333333 * rand);
	double tmp;
	if (rand <= -1e+91) {
		tmp = t_0;
	} else if (rand <= 1e+112) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = math.sqrt((a + -0.3333333333333333)) * (0.3333333333333333 * rand)
	tmp = 0
	if rand <= -1e+91:
		tmp = t_0
	elif rand <= 1e+112:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(sqrt(Float64(a + -0.3333333333333333)) * Float64(0.3333333333333333 * rand))
	tmp = 0.0
	if (rand <= -1e+91)
		tmp = t_0;
	elseif (rand <= 1e+112)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = sqrt((a + -0.3333333333333333)) * (0.3333333333333333 * rand);
	tmp = 0.0;
	if (rand <= -1e+91)
		tmp = t_0;
	elseif (rand <= 1e+112)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[Sqrt[N[(a + -0.3333333333333333), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 * rand), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1e+91], t$95$0, If[LessEqual[rand, 1e+112], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.00000000000000008e91 or 9.9999999999999993e111 < rand

   1. Initial program 98.7%

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

   3. Taylor expanded in rand around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -1.00000000000000008e91 < rand < 9.9999999999999993e111

   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. lower-+.f64 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 3: 91.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3333333333333333 \cdot \sqrt{a}, rand, -0.3333333333333333\right)\\ \mathbf{if}\;rand \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 6.5 \cdot 10^{+111}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (fma (* 0.3333333333333333 (sqrt a)) rand -0.3333333333333333)))
   (if (<= rand -1.15e+91)
     t_0
     (if (<= rand 6.5e+111) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = fma((0.3333333333333333 * sqrt(a)), rand, -0.3333333333333333);
	double tmp;
	if (rand <= -1.15e+91) {
		tmp = t_0;
	} else if (rand <= 6.5e+111) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, rand)
	t_0 = fma(Float64(0.3333333333333333 * sqrt(a)), rand, -0.3333333333333333)
	tmp = 0.0
	if (rand <= -1.15e+91)
		tmp = t_0;
	elseif (rand <= 6.5e+111)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * rand + -0.3333333333333333), $MachinePrecision]}, If[LessEqual[rand, -1.15e+91], t$95$0, If[LessEqual[rand, 6.5e+111], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.14999999999999996e91 or 6.5000000000000002e111 < rand

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.4%

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

   if -1.14999999999999996e91 < rand < 6.5000000000000002e111

   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. lower-+.f64 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 4: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 5: 63.4% 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 99.5%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
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. lower-+.f64 59.5

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

5. Applied rewrites 59.5%

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

Reproduce

herbie shell --seed 2024212 
(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))))