Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.8%

Time: 12.1s

Alternatives: 7

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 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. 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: 92.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -8 \cdot 10^{+77}:\\ \;\;\;\;\left(0.3333333333333333 \cdot rand\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 9 \cdot 10^{+73}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -8e+77)
   (* (* 0.3333333333333333 rand) (sqrt a))
   (if (<= rand 9e+73)
     (+ a -0.3333333333333333)
     (* rand (* 0.3333333333333333 (sqrt a))))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -8e+77) {
		tmp = (0.3333333333333333 * rand) * sqrt(a);
	} else if (rand <= 9e+73) {
		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 <= (-8d+77)) then
        tmp = (0.3333333333333333d0 * rand) * sqrt(a)
    else if (rand <= 9d+73) 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 <= -8e+77) {
		tmp = (0.3333333333333333 * rand) * Math.sqrt(a);
	} else if (rand <= 9e+73) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = rand * (0.3333333333333333 * Math.sqrt(a));
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -8e+77:
		tmp = (0.3333333333333333 * rand) * math.sqrt(a)
	elif rand <= 9e+73:
		tmp = a + -0.3333333333333333
	else:
		tmp = rand * (0.3333333333333333 * math.sqrt(a))
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -8e+77)
		tmp = Float64(Float64(0.3333333333333333 * rand) * sqrt(a));
	elseif (rand <= 9e+73)
		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 <= -8e+77)
		tmp = (0.3333333333333333 * rand) * sqrt(a);
	elseif (rand <= 9e+73)
		tmp = a + -0.3333333333333333;
	else
		tmp = rand * (0.3333333333333333 * sqrt(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -8e+77], N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 9e+73], 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 < -7.99999999999999986e77

   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. 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 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 87.9%

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

   if -7.99999999999999986e77 < rand < 8.99999999999999969e73

   1. Initial program 100.0%

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

   3. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if 8.99999999999999969e73 < 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in rand around inf

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

   10. Applied rewrites 89.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -8 \cdot 10^{+77}:\\ \;\;\;\;\left(0.3333333333333333 \cdot rand\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 9 \cdot 10^{+73}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))