Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 3.4s

Alternatives: 10

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(N[(a - 0.3333333333333333), $MachinePrecision] * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   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. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. sub-flip N/A

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

   6. lift--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (a rand)
 :precision binary64
 (fma
  (/ (- a 0.3333333333333333) (sqrt (fma 9.0 a -3.0)))
  rand
  (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	return fma(((a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, (a - 0.3333333333333333));
}

Julia

function code(a, rand)
	return fma(Float64(Float64(a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, Float64(a - 0.3333333333333333))
end

Wolfram

code[a_, rand_] := N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   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. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 1.5× speedup

Math

\[\left(\frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}} - -1\right) \cdot \left(a - 0.3333333333333333\right) \]

FPCore

(FPCore (a rand)
 :precision binary64
 (* (- (/ rand (sqrt (fma 9.0 a -3.0))) -1.0) (- a 0.3333333333333333)))

C

double code(double a, double rand) {
	return ((rand / sqrt(fma(9.0, a, -3.0))) - -1.0) * (a - 0.3333333333333333);
}

Julia

function code(a, rand)
	return Float64(Float64(Float64(rand / sqrt(fma(9.0, a, -3.0))) - -1.0) * Float64(a - 0.3333333333333333))
end

Wolfram

code[a_, rand_] := N[(N[(N[(rand / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   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 99.7%

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

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}} - -1\right) \cdot \left(a - 0.3333333333333333\right)} \]
4. Add Preprocessing

Alternative 4: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 3300000:\\ \;\;\;\;\frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}} \cdot \left(a - 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.1111111111111111 \cdot \sqrt{9 \cdot a}, rand, a\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= a 3300000.0)
   (* (/ rand (sqrt (fma 9.0 a -3.0))) (- a 0.3333333333333333))
   (fma (* 0.1111111111111111 (sqrt (* 9.0 a))) rand a)))

C

double code(double a, double rand) {
	double tmp;
	if (a <= 3300000.0) {
		tmp = (rand / sqrt(fma(9.0, a, -3.0))) * (a - 0.3333333333333333);
	} else {
		tmp = fma((0.1111111111111111 * sqrt((9.0 * a))), rand, a);
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= 3300000.0)
		tmp = Float64(Float64(rand / sqrt(fma(9.0, a, -3.0))) * Float64(a - 0.3333333333333333));
	else
		tmp = fma(Float64(0.1111111111111111 * sqrt(Float64(9.0 * a))), rand, a);
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, 3300000.0], N[(N[(rand / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(0.1111111111111111 * N[Sqrt[N[(9.0 * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.3e6

   1. Initial program 99.7%

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

   2. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}} \]
   3. 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 30.3%

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

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot \left(a - 0.3333333333333333\right)}}} \]
   5. 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

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

      4. mult-flip-rev N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 37.1%

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

   if 3.3e6 < a

   1. Initial program 99.7%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 97.3%

         \[\leadsto 0.1111111111111111 \cdot \left(a \cdot \left(9 + rand \cdot \sqrt{\frac{9}{a}}\right)\right) \]

   6. Applied rewrites 97.3%

      \[\leadsto \color{blue}{0.1111111111111111 \cdot \left(a \cdot \left(9 + rand \cdot \sqrt{\frac{9}{a}}\right)\right)} \]

   7. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 97.7%

         \[\leadsto a + 0.1111111111111111 \cdot \left(rand \cdot \sqrt{9 \cdot a}\right) \]

   9. Applied rewrites 97.7%

      \[\leadsto a + \color{blue}{0.1111111111111111 \cdot \left(rand \cdot \sqrt{9 \cdot a}\right)} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 97.7%

          \[\leadsto \mathsf{fma}\left(0.1111111111111111 \cdot \sqrt{9 \cdot a}, rand, a\right) \]

   11. Applied rewrites 97.7%

       \[\leadsto \mathsf{fma}\left(0.1111111111111111 \cdot \sqrt{9 \cdot a}, rand, a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   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. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in a around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{9}{a}}}}, rand, a - 0.3333333333333333\right) \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 98.9%

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

6. Applied rewrites 98.9%

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

Alternative 6: 98.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(a / N[Sqrt[N[(9.0 * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   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. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in a around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{9}{a}}}}, rand, a - 0.3333333333333333\right) \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 98.9%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 98.8%

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

9. Applied rewrites 98.8%

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

Alternative 7: 97.7% accurate, 2.2× speedup

Math

\[\mathsf{fma}\left(0.1111111111111111 \cdot \sqrt{9 \cdot a}, rand, a\right) \]

FPCore

(FPCore (a rand)
 :precision binary64
 (fma (* 0.1111111111111111 (sqrt (* 9.0 a))) rand a))

C

double code(double a, double rand) {
	return fma((0.1111111111111111 * sqrt((9.0 * a))), rand, a);
}

Julia

function code(a, rand)
	return fma(Float64(0.1111111111111111 * sqrt(Float64(9.0 * a))), rand, a)
end

Wolfram

code[a_, rand_] := N[(N[(0.1111111111111111 * N[Sqrt[N[(9.0 * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + a), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 97.3%

      \[\leadsto 0.1111111111111111 \cdot \left(a \cdot \left(9 + rand \cdot \sqrt{\frac{9}{a}}\right)\right) \]

6. Applied rewrites 97.3%

   \[\leadsto \color{blue}{0.1111111111111111 \cdot \left(a \cdot \left(9 + rand \cdot \sqrt{\frac{9}{a}}\right)\right)} \]

7. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 97.7%

      \[\leadsto a + 0.1111111111111111 \cdot \left(rand \cdot \sqrt{9 \cdot a}\right) \]

9. Applied rewrites 97.7%

   \[\leadsto a + \color{blue}{0.1111111111111111 \cdot \left(rand \cdot \sqrt{9 \cdot a}\right)} \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. lower-fma.f64 N/A

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

    8. lower-*.f64 97.7%

       \[\leadsto \mathsf{fma}\left(0.1111111111111111 \cdot \sqrt{9 \cdot a}, rand, a\right) \]

11. Applied rewrites 97.7%

    \[\leadsto \mathsf{fma}\left(0.1111111111111111 \cdot \sqrt{9 \cdot a}, rand, a\right) \]
12. Add Preprocessing

Alternative 8: 97.7% accurate, 2.2× speedup

Math

\[\mathsf{fma}\left(0.1111111111111111 \cdot rand, \sqrt{9 \cdot a}, a\right) \]

FPCore

(FPCore (a rand)
 :precision binary64
 (fma (* 0.1111111111111111 rand) (sqrt (* 9.0 a)) a))

C

double code(double a, double rand) {
	return fma((0.1111111111111111 * rand), sqrt((9.0 * a)), a);
}

Julia

function code(a, rand)
	return fma(Float64(0.1111111111111111 * rand), sqrt(Float64(9.0 * a)), a)
end

Wolfram

code[a_, rand_] := N[(N[(0.1111111111111111 * rand), $MachinePrecision] * N[Sqrt[N[(9.0 * a), $MachinePrecision]], $MachinePrecision] + a), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 97.3%

      \[\leadsto 0.1111111111111111 \cdot \left(a \cdot \left(9 + rand \cdot \sqrt{\frac{9}{a}}\right)\right) \]

6. Applied rewrites 97.3%

   \[\leadsto \color{blue}{0.1111111111111111 \cdot \left(a \cdot \left(9 + rand \cdot \sqrt{\frac{9}{a}}\right)\right)} \]

7. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 97.7%

      \[\leadsto a + 0.1111111111111111 \cdot \left(rand \cdot \sqrt{9 \cdot a}\right) \]

9. Applied rewrites 97.7%

   \[\leadsto a + \color{blue}{0.1111111111111111 \cdot \left(rand \cdot \sqrt{9 \cdot a}\right)} \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*r* N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-*.f64 97.7%

       \[\leadsto \mathsf{fma}\left(0.1111111111111111 \cdot rand, \sqrt{9 \cdot a}, a\right) \]

11. Applied rewrites 97.7%

    \[\leadsto \mathsf{fma}\left(0.1111111111111111 \cdot rand, \sqrt{9 \cdot a}, a\right) \]
12. Add Preprocessing

Alternative 9: 91.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{if}\;rand \leq -1.02 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\left(a - 0.3333333333333333\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (/ rand (sqrt (/ 9.0 a)))))
   (if (<= rand -1.02e+116)
     t_0
     (if (<= rand 1.05e+90) (* (- a 0.3333333333333333) 1.0) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand / sqrt((9.0 / a));
	double tmp;
	if (rand <= -1.02e+116) {
		tmp = t_0;
	} else if (rand <= 1.05e+90) {
		tmp = (a - 0.3333333333333333) * 1.0;
	} 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 <= (-1.02d+116)) then
        tmp = t_0
    else if (rand <= 1.05d+90) then
        tmp = (a - 0.3333333333333333d0) * 1.0d0
    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 <= -1.02e+116) {
		tmp = t_0;
	} else if (rand <= 1.05e+90) {
		tmp = (a - 0.3333333333333333) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand / math.sqrt((9.0 / a))
	tmp = 0
	if rand <= -1.02e+116:
		tmp = t_0
	elif rand <= 1.05e+90:
		tmp = (a - 0.3333333333333333) * 1.0
	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 <= -1.02e+116)
		tmp = t_0;
	elseif (rand <= 1.05e+90)
		tmp = Float64(Float64(a - 0.3333333333333333) * 1.0);
	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 <= -1.02e+116)
		tmp = t_0;
	elseif (rand <= 1.05e+90)
		tmp = (a - 0.3333333333333333) * 1.0;
	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, -1.02e+116], t$95$0, If[LessEqual[rand, 1.05e+90], N[(N[(a - 0.3333333333333333), $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -1.0199999999999999e116 or 1.0499999999999999e90 < rand

   1. Initial program 99.7%

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

   2. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}} \]
   3. 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 30.3%

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

   4. Applied rewrites 30.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]
   6. 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 36.3%

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

   7. Applied rewrites 36.3%

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

   if -1.0199999999999999e116 < rand < 1.0499999999999999e90

   1. Initial program 99.7%

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

   2. 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 63.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 63.8%

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

   6. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\left(a - 0.3333333333333333\right) \cdot 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.8% accurate, 4.5× speedup

Math

\[\left(a - 0.3333333333333333\right) \cdot 1 \]

FPCore

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

C

double code(double a, double rand) {
	return (a - 0.3333333333333333) * 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 = (a - 0.3333333333333333d0) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, rand_] := N[(N[(a - 0.3333333333333333), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. 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 63.8%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 63.8%

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

6. Applied rewrites 63.8%

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

Reproduce

herbie shell --seed 2025191 
(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))))