Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 2.9s

Alternatives: 1

Speedup: 1.2×

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, rand)
use fmin_fmax_functions
    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

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 99.8%

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-flip N/A

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

   10. distribute-lft-in N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 99.8%

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

3. Applied rewrites 99.8%

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

Reproduce

herbie shell --seed 2025192 
(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))))