Octave 3.8, oct_fill_randg

Percentage Accurate: 99.8% → 99.8%

Time: 3.7s

Alternatives: 11

Speedup: 1.4×

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: 99.8% 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, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\\ t\_0 \cdot \left(1 + \frac{1}{\sqrt{9 \cdot t\_0}} \cdot rand\right) \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (let* ((t_0
        (* (+ 1.0 (/ a -0.3333333333333333)) -0.3333333333333333)))
  (* t_0 (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 t_0))) rand)))))

C

double code(double a, double rand) {
	double t_0 = (1.0 + (a / -0.3333333333333333)) * -0.3333333333333333;
	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 = (1.0d0 + (a / (-0.3333333333333333d0))) * (-0.3333333333333333d0)
    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 = (1.0 + (a / -0.3333333333333333)) * -0.3333333333333333;
	return t_0 * (1.0 + ((1.0 / Math.sqrt((9.0 * t_0))) * rand));
}

Python

def code(a, rand):
	t_0 = (1.0 + (a / -0.3333333333333333)) * -0.3333333333333333
	return t_0 * (1.0 + ((1.0 / math.sqrt((9.0 * t_0))) * rand))

Julia

function code(a, rand)
	t_0 = Float64(Float64(1.0 + Float64(a / -0.3333333333333333)) * -0.3333333333333333)
	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 = (1.0 + (a / -0.3333333333333333)) * -0.3333333333333333;
	tmp = t_0 * (1.0 + ((1.0 / sqrt((9.0 * t_0))) * rand));
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[(1.0 + N[(a / -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $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]]

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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip 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. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

\[\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \left(\frac{rand}{\sqrt{\left(a - 0.3333333333333333\right) \cdot 9}} - -1\right) \]

FPCore

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

C

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

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
    code = ((1.0d0 + (a / (-0.3333333333333333d0))) * (-0.3333333333333333d0)) * ((rand / sqrt(((a - 0.3333333333333333d0) * 9.0d0))) - (-1.0d0))
end function

Java

public static double code(double a, double rand) {
	return ((1.0 + (a / -0.3333333333333333)) * -0.3333333333333333) * ((rand / Math.sqrt(((a - 0.3333333333333333) * 9.0))) - -1.0);
}

Python

def code(a, rand):
	return ((1.0 + (a / -0.3333333333333333)) * -0.3333333333333333) * ((rand / math.sqrt(((a - 0.3333333333333333) * 9.0))) - -1.0)

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = ((1.0 + (a / -0.3333333333333333)) * -0.3333333333333333) * ((rand / sqrt(((a - 0.3333333333333333) * 9.0))) - -1.0);
end

Wolfram

code[a_, rand_] := N[(N[(N[(1.0 + N[(a / -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(N[(rand / N[Sqrt[N[(N[(a - 0.3333333333333333), $MachinePrecision] * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip 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. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)}}} \cdot rand\right) \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.8% accurate, 1.4× speedup

Math

\[\left(\frac{rand}{\sqrt{\frac{9}{a}} \cdot a} - -1\right) \cdot \left(a - 0.3333333333333333\right) \]

FPCore

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

C

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

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
    code = ((rand / (sqrt((9.0d0 / a)) * a)) - (-1.0d0)) * (a - 0.3333333333333333d0)
end function

Java

public static double code(double a, double rand) {
	return ((rand / (Math.sqrt((9.0 / a)) * a)) - -1.0) * (a - 0.3333333333333333);
}

Python

def code(a, rand):
	return ((rand / (math.sqrt((9.0 / a)) * a)) - -1.0) * (a - 0.3333333333333333)

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = ((rand / (sqrt((9.0 / a)) * a)) - -1.0) * (a - 0.3333333333333333);
end

Wolfram

code[a_, rand_] := N[(N[(N[(rand / N[(N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - -1.0), $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. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 98.9%

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\left(\frac{rand}{\sqrt{\frac{9}{a}} \cdot a} - -1\right) \cdot \left(a - 0.3333333333333333\right)} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\left(a - 0.3333333333333333\right) - \frac{0.3333333333333333 - a}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \cdot rand \]

FPCore

(FPCore (a rand)
  :precision binary64
  (-
 (- a 0.3333333333333333)
 (*
  (/ (- 0.3333333333333333 a) (sqrt (* 9.0 (- a 0.3333333333333333))))
  rand)))

C

double code(double a, double rand) {
	return (a - 0.3333333333333333) - (((0.3333333333333333 - a) / sqrt((9.0 * (a - 0.3333333333333333)))) * 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
    code = (a - 0.3333333333333333d0) - (((0.3333333333333333d0 - a) / sqrt((9.0d0 * (a - 0.3333333333333333d0)))) * rand)
end function

Java

public static double code(double a, double rand) {
	return (a - 0.3333333333333333) - (((0.3333333333333333 - a) / Math.sqrt((9.0 * (a - 0.3333333333333333)))) * rand);
}

Python

def code(a, rand):
	return (a - 0.3333333333333333) - (((0.3333333333333333 - a) / math.sqrt((9.0 * (a - 0.3333333333333333)))) * rand)

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (a - 0.3333333333333333) - (((0.3333333333333333 - a) / sqrt((9.0 * (a - 0.3333333333333333)))) * rand);
end

Wolfram

code[a_, rand_] := N[(N[(a - 0.3333333333333333), $MachinePrecision] - N[(N[(N[(0.3333333333333333 - a), $MachinePrecision] / N[Sqrt[N[(9.0 * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip 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. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)}}} \cdot rand\right) \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

7. Applied rewrites 99.5%

   \[\leadsto \left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \color{blue}{\left(\frac{rand}{\sqrt{\left(a - 0.3333333333333333\right) \cdot 9}} - -1\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sum-to-mult-rev N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. sub-flip N/A

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

   9. lift--.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-flip N/A

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

   12. lift-/.f64 N/A

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

   13. mult-flip N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift--.f64 N/A

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

   18. *-commutative N/A

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

   19. metadata-eval N/A

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

   20. +-commutative N/A

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

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(a - 0.3333333333333333\right) - \frac{0.3333333333333333 - a}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \cdot rand} \]
10. Add Preprocessing

Alternative 5: 99.5% accurate, 1.5× speedup

Math

\[\left(a - 0.3333333333333333\right) - \left(0.3333333333333333 - a\right) \cdot \frac{rand}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (-
 (- a 0.3333333333333333)
 (*
  (- 0.3333333333333333 a)
  (/ rand (sqrt (* 9.0 (- a 0.3333333333333333)))))))

C

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

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
    code = (a - 0.3333333333333333d0) - ((0.3333333333333333d0 - a) * (rand / sqrt((9.0d0 * (a - 0.3333333333333333d0)))))
end function

Java

public static double code(double a, double rand) {
	return (a - 0.3333333333333333) - ((0.3333333333333333 - a) * (rand / Math.sqrt((9.0 * (a - 0.3333333333333333)))));
}

Python

def code(a, rand):
	return (a - 0.3333333333333333) - ((0.3333333333333333 - a) * (rand / math.sqrt((9.0 * (a - 0.3333333333333333)))))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (a - 0.3333333333333333) - ((0.3333333333333333 - a) * (rand / sqrt((9.0 * (a - 0.3333333333333333)))));
end

Wolfram

code[a_, rand_] := N[(N[(a - 0.3333333333333333), $MachinePrecision] - N[(N[(0.3333333333333333 - a), $MachinePrecision] * N[(rand / N[Sqrt[N[(9.0 * N[(a - 0.3333333333333333), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. add-flip-rev N/A

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

   6. lower--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-neg-out N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 99.8%

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

   14. lift-/.f64 N/A

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

   15. metadata-eval 99.8%

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

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(a - 0.3333333333333333\right) - \left(0.3333333333333333 - a\right) \cdot \frac{rand}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}}} \]
4. Add Preprocessing

Alternative 6: 98.9% accurate, 1.7× speedup

Math

\[\left(\frac{rand}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} - -1\right) \cdot \left(a - 0.3333333333333333\right) \]

FPCore

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

C

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

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
    code = ((rand / sqrt((9.0d0 * (a - 0.3333333333333333d0)))) - (-1.0d0)) * (a - 0.3333333333333333d0)
end function

Java

public static double code(double a, double rand) {
	return ((rand / Math.sqrt((9.0 * (a - 0.3333333333333333)))) - -1.0) * (a - 0.3333333333333333);
}

Python

def code(a, rand):
	return ((rand / math.sqrt((9.0 * (a - 0.3333333333333333)))) - -1.0) * (a - 0.3333333333333333)

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = ((rand / sqrt((9.0 * (a - 0.3333333333333333)))) - -1.0) * (a - 0.3333333333333333);
end

Wolfram

code[a_, rand_] := N[(N[(N[(rand / N[Sqrt[N[(9.0 * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. add-flip N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 99.8%

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. mult-flip-rev N/A

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

   13. lower-/.f64 99.8%

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

   14. lift-/.f64 N/A

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

   15. metadata-eval 99.8%

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

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\frac{rand}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} - -1\right) \cdot \left(a - 0.3333333333333333\right)} \]
4. Add Preprocessing

Alternative 7: 98.9% accurate, 1.8× speedup

Math

\[\left(\frac{rand}{\sqrt{9 \cdot a}} - -1\right) \cdot \left(a - 0.3333333333333333\right) \]

FPCore

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

C

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

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
    code = ((rand / sqrt((9.0d0 * a))) - (-1.0d0)) * (a - 0.3333333333333333d0)
end function

Java

public static double code(double a, double rand) {
	return ((rand / Math.sqrt((9.0 * a))) - -1.0) * (a - 0.3333333333333333);
}

Python

def code(a, rand):
	return ((rand / math.sqrt((9.0 * a))) - -1.0) * (a - 0.3333333333333333)

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = ((rand / sqrt((9.0 * a))) - -1.0) * (a - 0.3333333333333333);
end

Wolfram

code[a_, rand_] := N[(N[(N[(rand / N[Sqrt[N[(9.0 * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $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. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 98.9%

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in a around 0

   \[\leadsto \left(\frac{rand}{\sqrt{9 \cdot a}} - -1\right) \cdot \left(a - 0.3333333333333333\right) \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 98.9%

      \[\leadsto \left(\frac{rand}{\sqrt{9 \cdot a}} - -1\right) \cdot \left(a - 0.3333333333333333\right) \]

9. Applied rewrites 98.9%

   \[\leadsto \left(\frac{rand}{\sqrt{9 \cdot a}} - -1\right) \cdot \left(a - 0.3333333333333333\right) \]
10. Add Preprocessing

Alternative 8: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;rand \leq -1.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{elif}\;rand \leq 3.2 \cdot 10^{+128}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{a - 0.3333333333333333}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \cdot rand\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= rand -1.9e+59)
  (/ rand (sqrt (/ 9.0 a)))
  (if (<= rand 3.2e+128)
    (- a 0.3333333333333333)
    (*
     (/
      (- a 0.3333333333333333)
      (sqrt (* 9.0 (- a 0.3333333333333333))))
     rand))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -1.9e+59) {
		tmp = rand / sqrt((9.0 / a));
	} else if (rand <= 3.2e+128) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = ((a - 0.3333333333333333) / sqrt((9.0 * (a - 0.3333333333333333)))) * rand;
	}
	return tmp;
}

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) :: tmp
    if (rand <= (-1.9d+59)) then
        tmp = rand / sqrt((9.0d0 / a))
    else if (rand <= 3.2d+128) then
        tmp = a - 0.3333333333333333d0
    else
        tmp = ((a - 0.3333333333333333d0) / sqrt((9.0d0 * (a - 0.3333333333333333d0)))) * rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -1.9e+59) {
		tmp = rand / Math.sqrt((9.0 / a));
	} else if (rand <= 3.2e+128) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = ((a - 0.3333333333333333) / Math.sqrt((9.0 * (a - 0.3333333333333333)))) * rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -1.9e+59:
		tmp = rand / math.sqrt((9.0 / a))
	elif rand <= 3.2e+128:
		tmp = a - 0.3333333333333333
	else:
		tmp = ((a - 0.3333333333333333) / math.sqrt((9.0 * (a - 0.3333333333333333)))) * rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -1.9e+59)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	elseif (rand <= 3.2e+128)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(a - 0.3333333333333333) / sqrt(Float64(9.0 * Float64(a - 0.3333333333333333)))) * rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -1.9e+59)
		tmp = rand / sqrt((9.0 / a));
	elseif (rand <= 3.2e+128)
		tmp = a - 0.3333333333333333;
	else
		tmp = ((a - 0.3333333333333333) / sqrt((9.0 * (a - 0.3333333333333333)))) * rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -1.9e+59], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[rand, 3.2e+128], N[(a - 0.3333333333333333), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -1.9e59

   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. Taylor expanded in a around inf

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 30.5%

          \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \]

   7. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}}} \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      3. lower-/.f64 37.1%

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

   10. Applied rewrites 37.1%

       \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]

   if -1.9e59 < rand < 3.1999999999999999e128

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip 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. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.5%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.5%

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

   5. Applied rewrites 99.5%

      \[\leadsto \left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)}}} \cdot rand\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in rand around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 62.6%

         \[\leadsto -0.3333333333333333 \cdot \left(1 + -3 \cdot \color{blue}{a}\right) \]

   10. Applied rewrites 62.6%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(1 + -3 \cdot a\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. distribute-lft-in N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. sub-flip N/A

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

       13. lift--.f64 N/A

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

       14. *-lft-identity N/A

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

       15. metadata-eval 63.0%

           \[\leadsto a - 0.3333333333333333 \]

   12. Applied rewrites 63.0%

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

   if 3.1999999999999999e128 < rand

   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. Taylor expanded in a around inf

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 30.5%

          \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \]

   7. Applied rewrites 30.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 37.9%

         \[\leadsto \frac{a - 0.3333333333333333}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \cdot rand \]

   9. Applied rewrites 37.9%

      \[\leadsto \frac{a - 0.3333333333333333}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \cdot \color{blue}{rand} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{if}\;rand \leq -1.9 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 3.2 \cdot 10^{+128}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (let* ((t_0 (/ rand (sqrt (/ 9.0 a)))))
  (if (<= rand -1.9e+59)
    t_0
    (if (<= rand 3.2e+128) (- a 0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand / sqrt((9.0 / a));
	double tmp;
	if (rand <= -1.9e+59) {
		tmp = t_0;
	} else if (rand <= 3.2e+128) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = rand / sqrt((9.0d0 / a))
    if (rand <= (-1.9d+59)) then
        tmp = t_0
    else if (rand <= 3.2d+128) 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 / Math.sqrt((9.0 / a));
	double tmp;
	if (rand <= -1.9e+59) {
		tmp = t_0;
	} else if (rand <= 3.2e+128) {
		tmp = a - 0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand / math.sqrt((9.0 / a))
	tmp = 0
	if rand <= -1.9e+59:
		tmp = t_0
	elif rand <= 3.2e+128:
		tmp = a - 0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand / sqrt(Float64(9.0 / a)))
	tmp = 0.0
	if (rand <= -1.9e+59)
		tmp = t_0;
	elseif (rand <= 3.2e+128)
		tmp = Float64(a - 0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand / sqrt((9.0 / a));
	tmp = 0.0;
	if (rand <= -1.9e+59)
		tmp = t_0;
	elseif (rand <= 3.2e+128)
		tmp = a - 0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -1.9e+59], t$95$0, If[LessEqual[rand, 3.2e+128], N[(a - 0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.9e59 or 3.1999999999999999e128 < rand

   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. Taylor expanded in a around inf

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 30.5%

          \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}} \]

   7. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}}} \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      3. lower-/.f64 37.1%

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

   10. Applied rewrites 37.1%

       \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]

   if -1.9e59 < rand < 3.1999999999999999e128

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip 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. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.5%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.5%

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

   5. Applied rewrites 99.5%

      \[\leadsto \left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)}}} \cdot rand\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in rand around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 62.6%

         \[\leadsto -0.3333333333333333 \cdot \left(1 + -3 \cdot \color{blue}{a}\right) \]

   10. Applied rewrites 62.6%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(1 + -3 \cdot a\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. distribute-lft-in N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. sub-flip N/A

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

       13. lift--.f64 N/A

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

       14. *-lft-identity N/A

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

       15. metadata-eval 63.0%

           \[\leadsto a - 0.3333333333333333 \]

   12. Applied rewrites 63.0%

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

Alternative 10: 63.0% accurate, 3.4× speedup

Math

\[-0.3333333333333333 \cdot \left(1 + \frac{a}{-0.3333333333333333}\right) \]

FPCore

(FPCore (a rand)
  :precision binary64
  (* -0.3333333333333333 (+ 1.0 (/ a -0.3333333333333333))))

C

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

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
    code = (-0.3333333333333333d0) * (1.0d0 + (a / (-0.3333333333333333d0)))
end function

Java

public static double code(double a, double rand) {
	return -0.3333333333333333 * (1.0 + (a / -0.3333333333333333));
}

Python

def code(a, rand):
	return -0.3333333333333333 * (1.0 + (a / -0.3333333333333333))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = -0.3333333333333333 * (1.0 + (a / -0.3333333333333333));
end

Wolfram

code[a_, rand_] := N[(-0.3333333333333333 * N[(1.0 + N[(a / -0.3333333333333333), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip 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. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)}}} \cdot rand\right) \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in rand around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 62.6%

      \[\leadsto -0.3333333333333333 \cdot \left(1 + -3 \cdot \color{blue}{a}\right) \]

10. Applied rewrites 62.6%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(1 + -3 \cdot a\right)} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. metadata-eval N/A

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

    4. mult-flip N/A

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

    5. lower-/.f64 62.8%

       \[\leadsto -0.3333333333333333 \cdot \left(1 + \frac{a}{\color{blue}{-0.3333333333333333}}\right) \]

12. Applied rewrites 62.8%

    \[\leadsto -0.3333333333333333 \cdot \left(1 + \frac{a}{\color{blue}{-0.3333333333333333}}\right) \]
13. Add Preprocessing

Alternative 11: 62.8% accurate, 17.0× speedup

Math

\[a - 0.3333333333333333 \]

FPCore

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

C

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

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
    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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip 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. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
4. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-+.f64 N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \color{blue}{\left(\left(1 + \frac{a}{-0.3333333333333333}\right) \cdot -0.3333333333333333\right)}}} \cdot rand\right) \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in rand around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 62.6%

      \[\leadsto -0.3333333333333333 \cdot \left(1 + -3 \cdot \color{blue}{a}\right) \]

10. Applied rewrites 62.6%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(1 + -3 \cdot a\right)} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. distribute-rgt-in N/A

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

    5. *-commutative N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. metadata-eval N/A

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

    9. distribute-lft-in N/A

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. sub-flip N/A

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

    13. lift--.f64 N/A

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

    14. *-lft-identity N/A

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

    15. metadata-eval 63.0%

        \[\leadsto a - 0.3333333333333333 \]

12. Applied rewrites 63.0%

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

Reproduce

herbie shell --seed 2025258 
(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))))