Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 8.7s

Alternatives: 11

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(a - {3}^{-1}\right) \cdot \left(1 + \frac{rand}{-3} \cdot \frac{-1}{\sqrt{a - 0.3333333333333333}}\right) \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (*
  (- a (pow 3.0 -1.0))
  (+ 1.0 (* (/ rand -3.0) (/ -1.0 (sqrt (- a 0.3333333333333333)))))))

C

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

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a - (3.0d0 ** (-1.0d0))) * (1.0d0 + ((rand / (-3.0d0)) * ((-1.0d0) / sqrt((a - 0.3333333333333333d0)))))
end function

Java

public static double code(double a, double rand) {
	return (a - Math.pow(3.0, -1.0)) * (1.0 + ((rand / -3.0) * (-1.0 / Math.sqrt((a - 0.3333333333333333)))));
}

Python

def code(a, rand):
	return (a - math.pow(3.0, -1.0)) * (1.0 + ((rand / -3.0) * (-1.0 / math.sqrt((a - 0.3333333333333333)))))

Julia

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

MATLAB

function tmp = code(a, rand)
	tmp = (a - (3.0 ^ -1.0)) * (1.0 + ((rand / -3.0) * (-1.0 / sqrt((a - 0.3333333333333333)))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-neg-in N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lower-sqrt.f64 99.9

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

   18. lift-/.f64 N/A

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

   19. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \left(a - {3}^{-1}\right) \cdot \left(1 + \frac{rand}{-3} \cdot \frac{-1}{\sqrt{a - 0.3333333333333333}}\right) \]
6. Add Preprocessing

Alternative 2: 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. *-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. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft1-in N/A

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

   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 99.9%

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

Alternative 3: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(rand * N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * N[(a - 0.3333333333333333), $MachinePrecision]), $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. *-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. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft1-in N/A

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

   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 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{\left(a - 0.3333333333333333\right) \cdot 9}}, a - 0.3333333333333333, a - 0.3333333333333333\right)} \]
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. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 99.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 4: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(N[(rand / N[Sqrt[N[(a - 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 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 99.8%

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

Alternative 5: 92.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -3 \cdot 10^{+84} \lor \neg \left(rand \leq 4.7 \cdot 10^{+82}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.3333333333333333\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (or (<= rand -3e+84) (not (<= rand 4.7e+82)))
   (* (* (sqrt a) 0.3333333333333333) rand)
   (* (- a 0.3333333333333333) 1.0)))

C

double code(double a, double rand) {
	double tmp;
	if ((rand <= -3e+84) || !(rand <= 4.7e+82)) {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = (a - 0.3333333333333333) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if ((rand <= (-3d+84)) .or. (.not. (rand <= 4.7d+82))) then
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    else
        tmp = (a - 0.3333333333333333d0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if ((rand <= -3e+84) || !(rand <= 4.7e+82)) {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	} else {
		tmp = (a - 0.3333333333333333) * 1.0;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if (rand <= -3e+84) or not (rand <= 4.7e+82):
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	else:
		tmp = (a - 0.3333333333333333) * 1.0
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if ((rand <= -3e+84) || !(rand <= 4.7e+82))
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	else
		tmp = Float64(Float64(a - 0.3333333333333333) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if ((rand <= -3e+84) || ~((rand <= 4.7e+82)))
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	else
		tmp = (a - 0.3333333333333333) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[Or[LessEqual[rand, -3e+84], N[Not[LessEqual[rand, 4.7e+82]], $MachinePrecision]], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision], N[(N[(a - 0.3333333333333333), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if rand < -2.99999999999999996e84 or 4.7e82 < rand

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 90.1%

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

   if -2.99999999999999996e84 < rand < 4.7e82

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-neg-in N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 100.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in rand around 0

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

   7. Applied rewrites 95.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 95.5

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

   9. Applied rewrites 95.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -3 \cdot 10^{+84} \lor \neg \left(rand \leq 4.7 \cdot 10^{+82}\right):\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.3333333333333333\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;rand \leq -3 \cdot 10^{+84}:\\ \;\;\;\;\left(\sqrt{a} \cdot rand\right) \cdot 0.3333333333333333\\ \mathbf{elif}\;rand \leq 4.7 \cdot 10^{+82}:\\ \;\;\;\;\left(a - 0.3333333333333333\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{a} \cdot 0.3333333333333333\right) \cdot rand\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (if (<= rand -3e+84)
   (* (* (sqrt a) rand) 0.3333333333333333)
   (if (<= rand 4.7e+82)
     (* (- a 0.3333333333333333) 1.0)
     (* (* (sqrt a) 0.3333333333333333) rand))))

C

double code(double a, double rand) {
	double tmp;
	if (rand <= -3e+84) {
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	} else if (rand <= 4.7e+82) {
		tmp = (a - 0.3333333333333333) * 1.0;
	} else {
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: tmp
    if (rand <= (-3d+84)) then
        tmp = (sqrt(a) * rand) * 0.3333333333333333d0
    else if (rand <= 4.7d+82) then
        tmp = (a - 0.3333333333333333d0) * 1.0d0
    else
        tmp = (sqrt(a) * 0.3333333333333333d0) * rand
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double tmp;
	if (rand <= -3e+84) {
		tmp = (Math.sqrt(a) * rand) * 0.3333333333333333;
	} else if (rand <= 4.7e+82) {
		tmp = (a - 0.3333333333333333) * 1.0;
	} else {
		tmp = (Math.sqrt(a) * 0.3333333333333333) * rand;
	}
	return tmp;
}

Python

def code(a, rand):
	tmp = 0
	if rand <= -3e+84:
		tmp = (math.sqrt(a) * rand) * 0.3333333333333333
	elif rand <= 4.7e+82:
		tmp = (a - 0.3333333333333333) * 1.0
	else:
		tmp = (math.sqrt(a) * 0.3333333333333333) * rand
	return tmp

Julia

function code(a, rand)
	tmp = 0.0
	if (rand <= -3e+84)
		tmp = Float64(Float64(sqrt(a) * rand) * 0.3333333333333333);
	elseif (rand <= 4.7e+82)
		tmp = Float64(Float64(a - 0.3333333333333333) * 1.0);
	else
		tmp = Float64(Float64(sqrt(a) * 0.3333333333333333) * rand);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	tmp = 0.0;
	if (rand <= -3e+84)
		tmp = (sqrt(a) * rand) * 0.3333333333333333;
	elseif (rand <= 4.7e+82)
		tmp = (a - 0.3333333333333333) * 1.0;
	else
		tmp = (sqrt(a) * 0.3333333333333333) * rand;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := If[LessEqual[rand, -3e+84], N[(N[(N[Sqrt[a], $MachinePrecision] * rand), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[rand, 4.7e+82], N[(N[(a - 0.3333333333333333), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sqrt[a], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * rand), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if rand < -2.99999999999999996e84

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      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 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.4%

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

   if -2.99999999999999996e84 < rand < 4.7e82

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-neg-in N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 100.0

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

      18. lift-/.f64 N/A

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

      19. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in rand around 0

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

   7. Applied rewrites 95.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 95.5

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

   9. Applied rewrites 95.5%

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

   if 4.7e82 < 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. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      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 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in rand around inf

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 91.4%

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

Alternative 7: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   4. +-commutative N/A

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

   5. distribute-lft1-in N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in rand around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower--.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 8: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, rand_] := N[(N[(0.3333333333333333 * rand), $MachinePrecision] * N[Sqrt[N[(a - 0.3333333333333333), $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. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 9: 97.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. *-commutative N/A

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

   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

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 97.1%

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

Alternative 10: 62.3% accurate, 7.6× speedup

Math

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

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

real(8) function code(a, rand)
    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 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-neg-in N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lower-sqrt.f64 99.9

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

   18. lift-/.f64 N/A

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

   19. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in rand around 0

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

7. Applied rewrites 63.0%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 63.0

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

9. Applied rewrites 63.0%

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

Alternative 11: 1.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot 1 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (-0.3333333333333333d0) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-neg-in N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lower-sqrt.f64 99.9

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

   18. lift-/.f64 N/A

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

   19. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in rand around 0

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

7. Applied rewrites 63.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 1.5%

    \[\leadsto \color{blue}{-0.3333333333333333} \cdot 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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))))