Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 6.1s

Alternatives: 10

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

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

FPCore

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

C

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

Fortran

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, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. sqrt-prod N/A

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

   8. metadata-eval N/A

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

   9. lift-sqrt.f64 N/A

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

   10. times-frac N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in rand around 0

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

   1. lower-*.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, \frac{rand}{\sqrt{a - 0.3333333333333333}}, 1\right) \cdot \left(a - 0.3333333333333333\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(0.3333333333333333 * N[(rand / N[Sqrt[N[(a - 0.3333333333333333), $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. Add Preprocessing
3. 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. Applied rewrites 99.7%

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

Alternative 4: 91.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -7.5 \cdot 10^{+83} \lor \neg \left(rand \leq 3.6 \cdot 10^{+62}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, -1, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -7.5e+83) (not (<= rand 3.6e+62)))
   (* (* (sqrt a) rand) 0.3333333333333333)
   (fma 0.3333333333333333 -1.0 a)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -7.5e+83) || !(rand <= 3.6e+62)) {
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	} else {
		tmp = fma(0.3333333333333333, -1.0, a);
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -7.5e+83) || !(rand <= 3.6e+62))
		tmp = Float64(Float64(sqrt(a) * rand) * 0.3333333333333333);
	else
		tmp = fma(0.3333333333333333, -1.0, a);
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -7.5e+83], N[Not[LessEqual[rand, 3.6e+62]], $MachinePrecision]], N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 * -1.0 + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -7.49999999999999989e83 or 3.6e62 < rand

   1. Initial program 99.5%

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

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. sqrt-prod N/A

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

      8. metadata-eval N/A

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

      9. lift-sqrt.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 97.4

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 87.9%

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

   if -7.49999999999999989e83 < rand < 3.6e62

   1. Initial program 99.9%

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

   3. Taylor expanded in rand around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in rand around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{3}, -1, a\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.4%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, -1, a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -7.5 \cdot 10^{+83} \lor \neg \left(rand \leq 3.6 \cdot 10^{+62}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, -1, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(\sqrt{a - 0.3333333333333333}, rand, -1\right), a\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-out-- N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 6: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
2. Add Preprocessing
3. 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. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 98.9

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

7. Applied rewrites 98.9%

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

Alternative 7: 98.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(\sqrt{a}, rand, -1\right), a\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-out-- N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 98.6%

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

Alternative 8: 97.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-out-- N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 96.9%

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

Alternative 9: 62.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333, -1, a\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in rand around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-out-- N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in rand around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3}, -1, a\right) \]
7. Step-by-step derivation

8. Applied rewrites 62.5%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, -1, a\right) \]
9. Add Preprocessing

Alternative 10: 61.9% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot a \end{array} \]

FPCore

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

C

double code(double a, double rand) {
	return 1.0 * a;
}

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. sqrt-prod N/A

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

   8. metadata-eval N/A

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

   9. lift-sqrt.f64 N/A

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

   10. times-frac N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 97.1

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

9. Applied rewrites 97.1%

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

10. Taylor expanded in rand around 0

    \[\leadsto 1 \cdot a \]
11. Step-by-step derivation

12. Applied rewrites 60.8%

    \[\leadsto 1 \cdot a \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025009 
(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))))