Octave 3.8, oct_fill_randg

Percentage Accurate: 25.0% → 69.1%

Time: 2.6s

Alternatives: 15

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{9}{a}}\\ t_1 := rand \cdot \left(a - 0.3333333333333333\right)\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+157}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\sqrt{\mathsf{fma}\left(9, a, -3\right) \cdot \mathsf{fma}\left(9, a, -3\right)}}}\\ \mathbf{elif}\;a \leq -0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, 0.1111111111111111\right)\right)}} \cdot 1\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{{a}^{2}}{a - -0.3333333333333333} \cdot 1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{t\_1}{a \cdot \left(\left(t\_0 + -1 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{1}{{a}^{2} \cdot {t\_0}^{5}}, 1.125 \cdot \frac{1}{{t\_0}^{3}}\right)}{{a}^{4}}\right) - \frac{1.5}{{a}^{2} \cdot t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (let* ((t_0 (sqrt (/ 9.0 a))) (t_1 (* rand (- a 0.3333333333333333))))
  (if (<= a -4.1e+157)
    (/ t_1 (sqrt (sqrt (* (fma 9.0 a -3.0) (fma 9.0 a -3.0)))))
    (if (<= a -0.33)
      (*
       (/
        (fma a a -0.1111111111111111)
        (/
         (fma (* a a) a 0.037037037037037035)
         (fma a a (fma -0.3333333333333333 a 0.1111111111111111))))
       1.0)
      (if (<= a 1.55e-162)
        (* (/ (pow a 2.0) (- a -0.3333333333333333)) 1.0)
        (if (<= a 3.2e+26)
          (/
           t_1
           (*
            a
            (-
             (+
              t_0
              (*
               -1.0
               (/
                (fma
                 1.6875
                 (/ 1.0 (* (pow a 2.0) (pow t_0 5.0)))
                 (* 1.125 (/ 1.0 (pow t_0 3.0))))
                (pow a 4.0))))
             (/ 1.5 (* (pow a 2.0) t_0)))))
          (fma
           (/ (- a 0.3333333333333333) (sqrt (fma 9.0 a -3.0)))
           rand
           (- a 0.3333333333333333))))))))

C

double code(double a, double rand) {
	double t_0 = sqrt((9.0 / a));
	double t_1 = rand * (a - 0.3333333333333333);
	double tmp;
	if (a <= -4.1e+157) {
		tmp = t_1 / sqrt(sqrt((fma(9.0, a, -3.0) * fma(9.0, a, -3.0))));
	} else if (a <= -0.33) {
		tmp = (fma(a, a, -0.1111111111111111) / (fma((a * a), a, 0.037037037037037035) / fma(a, a, fma(-0.3333333333333333, a, 0.1111111111111111)))) * 1.0;
	} else if (a <= 1.55e-162) {
		tmp = (pow(a, 2.0) / (a - -0.3333333333333333)) * 1.0;
	} else if (a <= 3.2e+26) {
		tmp = t_1 / (a * ((t_0 + (-1.0 * (fma(1.6875, (1.0 / (pow(a, 2.0) * pow(t_0, 5.0))), (1.125 * (1.0 / pow(t_0, 3.0)))) / pow(a, 4.0)))) - (1.5 / (pow(a, 2.0) * t_0))));
	} else {
		tmp = fma(((a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	t_0 = sqrt(Float64(9.0 / a))
	t_1 = Float64(rand * Float64(a - 0.3333333333333333))
	tmp = 0.0
	if (a <= -4.1e+157)
		tmp = Float64(t_1 / sqrt(sqrt(Float64(fma(9.0, a, -3.0) * fma(9.0, a, -3.0)))));
	elseif (a <= -0.33)
		tmp = Float64(Float64(fma(a, a, -0.1111111111111111) / Float64(fma(Float64(a * a), a, 0.037037037037037035) / fma(a, a, fma(-0.3333333333333333, a, 0.1111111111111111)))) * 1.0);
	elseif (a <= 1.55e-162)
		tmp = Float64(Float64((a ^ 2.0) / Float64(a - -0.3333333333333333)) * 1.0);
	elseif (a <= 3.2e+26)
		tmp = Float64(t_1 / Float64(a * Float64(Float64(t_0 + Float64(-1.0 * Float64(fma(1.6875, Float64(1.0 / Float64((a ^ 2.0) * (t_0 ^ 5.0))), Float64(1.125 * Float64(1.0 / (t_0 ^ 3.0)))) / (a ^ 4.0)))) - Float64(1.5 / Float64((a ^ 2.0) * t_0)))));
	else
		tmp = fma(Float64(Float64(a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(rand * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.1e+157], N[(t$95$1 / N[Sqrt[N[Sqrt[N[(N[(9.0 * a + -3.0), $MachinePrecision] * N[(9.0 * a + -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -0.33], N[(N[(N[(a * a + -0.1111111111111111), $MachinePrecision] / N[(N[(N[(a * a), $MachinePrecision] * a + 0.037037037037037035), $MachinePrecision] / N[(a * a + N[(-0.3333333333333333 * a + 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[a, 1.55e-162], N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[(a - -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[a, 3.2e+26], N[(t$95$1 / N[(a * N[(N[(t$95$0 + N[(-1.0 * N[(N[(1.6875 * N[(1.0 / N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[t$95$0, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.125 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.5 / N[(N[Power[a, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -4.1000000000000002e157

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 14.8%

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

   8. Applied rewrites 14.8%

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

   if -4.1000000000000002e157 < a < -0.33000000000000002

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. remove-sound-/ N/A

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

      4. lower-/.f64 N/A

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

      5. sub-flip N/A

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

      6. cube-neg N/A

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

      7. metadata-eval N/A

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

      8. unpow3 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \color{blue}{\frac{1}{27}}\right)}{a \cdot a + \left(\frac{-1}{3} \cdot \frac{-1}{3} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\color{blue}{\mathsf{fma}\left(a, a, \frac{-1}{3} \cdot \frac{-1}{3} + a \cdot \frac{-1}{3}\right)}}} \cdot 1 \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{1}{9}} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{\frac{1}{3}}{3}} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{a \cdot \frac{-1}{3} + \frac{\frac{1}{3}}{3}}\right)}} \cdot 1 \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{-1}{3} \cdot a} + \frac{\frac{1}{3}}{3}\right)}} \cdot 1 \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, a, \frac{\frac{1}{3}}{3}\right)}\right)}} \cdot 1 \]

      18. metadata-eval 10.3%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, \color{blue}{0.1111111111111111}\right)\right)}} \cdot 1 \]

   8. Applied rewrites 10.3%

      \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, 0.1111111111111111\right)\right)}}} \cdot 1 \]

   if -0.33000000000000002 < a < 1.5499999999999999e-162

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-pow.f64 35.7%

         \[\leadsto \frac{{a}^{\color{blue}{2}}}{a - -0.3333333333333333} \cdot 1 \]

   9. Applied rewrites 35.7%

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]

   if 1.5499999999999999e-162 < a < 3.2000000000000003e26

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

      \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{a \cdot \color{blue}{\left(\left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\frac{27}{16} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{5}} + \frac{9}{8} \cdot \frac{1}{{\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{4}}\right) - \frac{\frac{3}{2}}{{a}^{2} \cdot \sqrt{\frac{9}{a}}}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\frac{27}{16} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{5}} + \frac{9}{8} \cdot \frac{1}{{\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{4}}\right) - \color{blue}{\frac{\frac{3}{2}}{{a}^{2} \cdot \sqrt{\frac{9}{a}}}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\frac{27}{16} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{5}} + \frac{9}{8} \cdot \frac{1}{{\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{4}}\right) - \frac{\frac{3}{2}}{\color{blue}{{a}^{2} \cdot \sqrt{\frac{9}{a}}}}\right)} \]

   9. Applied rewrites 12.3%

      \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{a \cdot \color{blue}{\left(\left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{5}}, 1.125 \cdot \frac{1}{{\left(\sqrt{\frac{9}{a}}\right)}^{3}}\right)}{{a}^{4}}\right) - \frac{1.5}{{a}^{2} \cdot \sqrt{\frac{9}{a}}}\right)}} \]

   if 3.2000000000000003e26 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 65.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := rand \cdot \left(a - 0.3333333333333333\right)\\ t_1 := \sqrt{\frac{9}{a}}\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+157}:\\ \;\;\;\;\frac{t\_0}{\sqrt{\sqrt{\mathsf{fma}\left(9, a, -3\right) \cdot \mathsf{fma}\left(9, a, -3\right)}}}\\ \mathbf{elif}\;a \leq -0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, 0.1111111111111111\right)\right)}} \cdot 1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-205}:\\ \;\;\;\;\frac{{a}^{2}}{a - -0.3333333333333333} \cdot 1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{t\_0}{a \cdot \left(t\_1 + -1 \cdot \frac{\mathsf{fma}\left(1.5, \frac{1}{t\_1}, 1.125 \cdot \frac{1}{{a}^{2} \cdot {t\_1}^{3}}\right)}{{a}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (let* ((t_0 (* rand (- a 0.3333333333333333))) (t_1 (sqrt (/ 9.0 a))))
  (if (<= a -4.1e+157)
    (/ t_0 (sqrt (sqrt (* (fma 9.0 a -3.0) (fma 9.0 a -3.0)))))
    (if (<= a -0.33)
      (*
       (/
        (fma a a -0.1111111111111111)
        (/
         (fma (* a a) a 0.037037037037037035)
         (fma a a (fma -0.3333333333333333 a 0.1111111111111111))))
       1.0)
      (if (<= a 2e-205)
        (* (/ (pow a 2.0) (- a -0.3333333333333333)) 1.0)
        (if (<= a 3.2e+26)
          (/
           t_0
           (*
            a
            (+
             t_1
             (*
              -1.0
              (/
               (fma
                1.5
                (/ 1.0 t_1)
                (* 1.125 (/ 1.0 (* (pow a 2.0) (pow t_1 3.0)))))
               (pow a 2.0))))))
          (fma
           (/ (- a 0.3333333333333333) (sqrt (fma 9.0 a -3.0)))
           rand
           (- a 0.3333333333333333))))))))

C

double code(double a, double rand) {
	double t_0 = rand * (a - 0.3333333333333333);
	double t_1 = sqrt((9.0 / a));
	double tmp;
	if (a <= -4.1e+157) {
		tmp = t_0 / sqrt(sqrt((fma(9.0, a, -3.0) * fma(9.0, a, -3.0))));
	} else if (a <= -0.33) {
		tmp = (fma(a, a, -0.1111111111111111) / (fma((a * a), a, 0.037037037037037035) / fma(a, a, fma(-0.3333333333333333, a, 0.1111111111111111)))) * 1.0;
	} else if (a <= 2e-205) {
		tmp = (pow(a, 2.0) / (a - -0.3333333333333333)) * 1.0;
	} else if (a <= 3.2e+26) {
		tmp = t_0 / (a * (t_1 + (-1.0 * (fma(1.5, (1.0 / t_1), (1.125 * (1.0 / (pow(a, 2.0) * pow(t_1, 3.0))))) / pow(a, 2.0)))));
	} else {
		tmp = fma(((a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(a - 0.3333333333333333))
	t_1 = sqrt(Float64(9.0 / a))
	tmp = 0.0
	if (a <= -4.1e+157)
		tmp = Float64(t_0 / sqrt(sqrt(Float64(fma(9.0, a, -3.0) * fma(9.0, a, -3.0)))));
	elseif (a <= -0.33)
		tmp = Float64(Float64(fma(a, a, -0.1111111111111111) / Float64(fma(Float64(a * a), a, 0.037037037037037035) / fma(a, a, fma(-0.3333333333333333, a, 0.1111111111111111)))) * 1.0);
	elseif (a <= 2e-205)
		tmp = Float64(Float64((a ^ 2.0) / Float64(a - -0.3333333333333333)) * 1.0);
	elseif (a <= 3.2e+26)
		tmp = Float64(t_0 / Float64(a * Float64(t_1 + Float64(-1.0 * Float64(fma(1.5, Float64(1.0 / t_1), Float64(1.125 * Float64(1.0 / Float64((a ^ 2.0) * (t_1 ^ 3.0))))) / (a ^ 2.0))))));
	else
		tmp = fma(Float64(Float64(a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -4.1e+157], N[(t$95$0 / N[Sqrt[N[Sqrt[N[(N[(9.0 * a + -3.0), $MachinePrecision] * N[(9.0 * a + -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -0.33], N[(N[(N[(a * a + -0.1111111111111111), $MachinePrecision] / N[(N[(N[(a * a), $MachinePrecision] * a + 0.037037037037037035), $MachinePrecision] / N[(a * a + N[(-0.3333333333333333 * a + 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[a, 2e-205], N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[(a - -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[a, 3.2e+26], N[(t$95$0 / N[(a * N[(t$95$1 + N[(-1.0 * N[(N[(1.5 * N[(1.0 / t$95$1), $MachinePrecision] + N[(1.125 * N[(1.0 / N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -4.1000000000000002e157

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 14.8%

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

   8. Applied rewrites 14.8%

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

   if -4.1000000000000002e157 < a < -0.33000000000000002

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. remove-sound-/ N/A

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

      4. lower-/.f64 N/A

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

      5. sub-flip N/A

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

      6. cube-neg N/A

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

      7. metadata-eval N/A

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

      8. unpow3 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \color{blue}{\frac{1}{27}}\right)}{a \cdot a + \left(\frac{-1}{3} \cdot \frac{-1}{3} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\color{blue}{\mathsf{fma}\left(a, a, \frac{-1}{3} \cdot \frac{-1}{3} + a \cdot \frac{-1}{3}\right)}}} \cdot 1 \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{1}{9}} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{\frac{1}{3}}{3}} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{a \cdot \frac{-1}{3} + \frac{\frac{1}{3}}{3}}\right)}} \cdot 1 \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{-1}{3} \cdot a} + \frac{\frac{1}{3}}{3}\right)}} \cdot 1 \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, a, \frac{\frac{1}{3}}{3}\right)}\right)}} \cdot 1 \]

      18. metadata-eval 10.3%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, \color{blue}{0.1111111111111111}\right)\right)}} \cdot 1 \]

   8. Applied rewrites 10.3%

      \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, 0.1111111111111111\right)\right)}}} \cdot 1 \]

   if -0.33000000000000002 < a < 2e-205

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-pow.f64 35.7%

         \[\leadsto \frac{{a}^{\color{blue}{2}}}{a - -0.3333333333333333} \cdot 1 \]

   9. Applied rewrites 35.7%

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]

   if 2e-205 < a < 3.2000000000000003e26

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

      \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{a \cdot \color{blue}{\left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\frac{3}{2} \cdot \frac{1}{\sqrt{\frac{9}{a}}} + \frac{9}{8} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{2}}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\sqrt{\frac{9}{a}} + \color{blue}{-1 \cdot \frac{\frac{3}{2} \cdot \frac{1}{\sqrt{\frac{9}{a}}} + \frac{9}{8} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{2}}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\sqrt{\frac{9}{a}} + -1 \cdot \color{blue}{\frac{\frac{3}{2} \cdot \frac{1}{\sqrt{\frac{9}{a}}} + \frac{9}{8} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{2}}}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\color{blue}{\frac{3}{2} \cdot \frac{1}{\sqrt{\frac{9}{a}}} + \frac{9}{8} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}}}{{a}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\color{blue}{\frac{3}{2} \cdot \frac{1}{\sqrt{\frac{9}{a}}}} + \frac{9}{8} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\frac{3}{2} \cdot \frac{1}{\sqrt{\frac{9}{a}}} + \frac{9}{8} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{\color{blue}{{a}^{2}}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{a \cdot \left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\frac{3}{2} \cdot \frac{1}{\sqrt{\frac{9}{a}}} + \frac{9}{8} \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}}{{a}^{\color{blue}{2}}}\right)} \]

   9. Applied rewrites 13.9%

      \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{a \cdot \color{blue}{\left(\sqrt{\frac{9}{a}} + -1 \cdot \frac{\mathsf{fma}\left(1.5, \frac{1}{\sqrt{\frac{9}{a}}}, 1.125 \cdot \frac{1}{{a}^{2} \cdot {\left(\sqrt{\frac{9}{a}}\right)}^{3}}\right)}{{a}^{2}}\right)}} \]

   if 3.2000000000000003e26 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+157}:\\ \;\;\;\;\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{\sqrt{\mathsf{fma}\left(9, a, -3\right) \cdot \mathsf{fma}\left(9, a, -3\right)}}}\\ \mathbf{elif}\;a \leq -0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, 0.1111111111111111\right)\right)}} \cdot 1\\ \mathbf{elif}\;a \leq 0.35:\\ \;\;\;\;\frac{{a}^{2}}{a - -0.3333333333333333} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a -4.1e+157)
  (/
   (* rand (- a 0.3333333333333333))
   (sqrt (sqrt (* (fma 9.0 a -3.0) (fma 9.0 a -3.0)))))
  (if (<= a -0.33)
    (*
     (/
      (fma a a -0.1111111111111111)
      (/
       (fma (* a a) a 0.037037037037037035)
       (fma a a (fma -0.3333333333333333 a 0.1111111111111111))))
     1.0)
    (if (<= a 0.35)
      (* (/ (pow a 2.0) (- a -0.3333333333333333)) 1.0)
      (fma
       (/ (- a 0.3333333333333333) (sqrt (fma 9.0 a -3.0)))
       rand
       (- a 0.3333333333333333))))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= -4.1e+157) {
		tmp = (rand * (a - 0.3333333333333333)) / sqrt(sqrt((fma(9.0, a, -3.0) * fma(9.0, a, -3.0))));
	} else if (a <= -0.33) {
		tmp = (fma(a, a, -0.1111111111111111) / (fma((a * a), a, 0.037037037037037035) / fma(a, a, fma(-0.3333333333333333, a, 0.1111111111111111)))) * 1.0;
	} else if (a <= 0.35) {
		tmp = (pow(a, 2.0) / (a - -0.3333333333333333)) * 1.0;
	} else {
		tmp = fma(((a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= -4.1e+157)
		tmp = Float64(Float64(rand * Float64(a - 0.3333333333333333)) / sqrt(sqrt(Float64(fma(9.0, a, -3.0) * fma(9.0, a, -3.0)))));
	elseif (a <= -0.33)
		tmp = Float64(Float64(fma(a, a, -0.1111111111111111) / Float64(fma(Float64(a * a), a, 0.037037037037037035) / fma(a, a, fma(-0.3333333333333333, a, 0.1111111111111111)))) * 1.0);
	elseif (a <= 0.35)
		tmp = Float64(Float64((a ^ 2.0) / Float64(a - -0.3333333333333333)) * 1.0);
	else
		tmp = fma(Float64(Float64(a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, -4.1e+157], N[(N[(rand * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sqrt[N[(N[(9.0 * a + -3.0), $MachinePrecision] * N[(9.0 * a + -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -0.33], N[(N[(N[(a * a + -0.1111111111111111), $MachinePrecision] / N[(N[(N[(a * a), $MachinePrecision] * a + 0.037037037037037035), $MachinePrecision] / N[(a * a + N[(-0.3333333333333333 * a + 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[a, 0.35], N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[(a - -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -4.1000000000000002e157

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 14.8%

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

   8. Applied rewrites 14.8%

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

   if -4.1000000000000002e157 < a < -0.33000000000000002

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. remove-sound-/ N/A

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

      4. lower-/.f64 N/A

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

      5. sub-flip N/A

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

      6. cube-neg N/A

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

      7. metadata-eval N/A

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

      8. unpow3 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \color{blue}{\frac{1}{27}}\right)}{a \cdot a + \left(\frac{-1}{3} \cdot \frac{-1}{3} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\color{blue}{\mathsf{fma}\left(a, a, \frac{-1}{3} \cdot \frac{-1}{3} + a \cdot \frac{-1}{3}\right)}}} \cdot 1 \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{1}{9}} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{\frac{1}{3}}{3}} + a \cdot \frac{-1}{3}\right)}} \cdot 1 \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{a \cdot \frac{-1}{3} + \frac{\frac{1}{3}}{3}}\right)}} \cdot 1 \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\frac{-1}{3} \cdot a} + \frac{\frac{1}{3}}{3}\right)}} \cdot 1 \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, \frac{1}{27}\right)}{\mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, a, \frac{\frac{1}{3}}{3}\right)}\right)}} \cdot 1 \]

      18. metadata-eval 10.3%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, \color{blue}{0.1111111111111111}\right)\right)}} \cdot 1 \]

   8. Applied rewrites 10.3%

      \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot a, a, 0.037037037037037035\right)}{\mathsf{fma}\left(a, a, \mathsf{fma}\left(-0.3333333333333333, a, 0.1111111111111111\right)\right)}}} \cdot 1 \]

   if -0.33000000000000002 < a < 0.34999999999999998

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-pow.f64 35.7%

         \[\leadsto \frac{{a}^{\color{blue}{2}}}{a - -0.3333333333333333} \cdot 1 \]

   9. Applied rewrites 35.7%

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]

   if 0.34999999999999998 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

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

Alternative 4: 59.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{\sqrt{\mathsf{fma}\left(9, a, -3\right) \cdot \mathsf{fma}\left(9, a, -3\right)}}}\\ \mathbf{elif}\;a \leq 0.35:\\ \;\;\;\;\frac{{a}^{2}}{a - -0.3333333333333333} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a -3.2e-69)
  (/
   (* rand (- a 0.3333333333333333))
   (sqrt (sqrt (* (fma 9.0 a -3.0) (fma 9.0 a -3.0)))))
  (if (<= a 0.35)
    (* (/ (pow a 2.0) (- a -0.3333333333333333)) 1.0)
    (fma
     (/ (- a 0.3333333333333333) (sqrt (fma 9.0 a -3.0)))
     rand
     (- a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= -3.2e-69) {
		tmp = (rand * (a - 0.3333333333333333)) / sqrt(sqrt((fma(9.0, a, -3.0) * fma(9.0, a, -3.0))));
	} else if (a <= 0.35) {
		tmp = (pow(a, 2.0) / (a - -0.3333333333333333)) * 1.0;
	} else {
		tmp = fma(((a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= -3.2e-69)
		tmp = Float64(Float64(rand * Float64(a - 0.3333333333333333)) / sqrt(sqrt(Float64(fma(9.0, a, -3.0) * fma(9.0, a, -3.0)))));
	elseif (a <= 0.35)
		tmp = Float64(Float64((a ^ 2.0) / Float64(a - -0.3333333333333333)) * 1.0);
	else
		tmp = fma(Float64(Float64(a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, -3.2e-69], N[(N[(rand * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sqrt[N[(N[(9.0 * a + -3.0), $MachinePrecision] * N[(9.0 * a + -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.35], N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[(a - -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.2e-69

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 14.8%

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

   8. Applied rewrites 14.8%

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

   if -3.2e-69 < a < 0.34999999999999998

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-pow.f64 35.7%

         \[\leadsto \frac{{a}^{\color{blue}{2}}}{a - -0.3333333333333333} \cdot 1 \]

   9. Applied rewrites 35.7%

      \[\leadsto \frac{\color{blue}{{a}^{2}}}{a - -0.3333333333333333} \cdot 1 \]

   if 0.34999999999999998 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

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

Alternative 5: 37.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{\sqrt{\mathsf{fma}\left(9, a, -3\right) \cdot \mathsf{fma}\left(9, a, -3\right)}}}\\ \mathbf{elif}\;a \leq 10000:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a -2e-310)
  (/
   (* rand (- a 0.3333333333333333))
   (sqrt (sqrt (* (fma 9.0 a -3.0) (fma 9.0 a -3.0)))))
  (if (<= a 10000.0)
    (/ rand (sqrt (/ 9.0 a)))
    (fma
     (/ (- a 0.3333333333333333) (sqrt (fma 9.0 a -3.0)))
     rand
     (- a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= -2e-310) {
		tmp = (rand * (a - 0.3333333333333333)) / sqrt(sqrt((fma(9.0, a, -3.0) * fma(9.0, a, -3.0))));
	} else if (a <= 10000.0) {
		tmp = rand / sqrt((9.0 / a));
	} else {
		tmp = fma(((a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= -2e-310)
		tmp = Float64(Float64(rand * Float64(a - 0.3333333333333333)) / sqrt(sqrt(Float64(fma(9.0, a, -3.0) * fma(9.0, a, -3.0)))));
	elseif (a <= 10000.0)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	else
		tmp = fma(Float64(Float64(a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, -2e-310], N[(N[(rand * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sqrt[N[(N[(9.0 * a + -3.0), $MachinePrecision] * N[(9.0 * a + -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 10000.0], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9999999999999939e-310

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. add-flip N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-fabs-rev N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-*.f64 14.8%

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

   8. Applied rewrites 14.8%

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

   if -1.9999999999999939e-310 < a < 1e4

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.2%

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

   9. Applied rewrites 13.2%

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

   if 1e4 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

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

Alternative 6: 32.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \frac{rand}{\sqrt{\left|0.3333333333333333 - a\right|}}\\ \mathbf{elif}\;a \leq 10000:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a -2e-310)
  (*
   (fma 0.3333333333333333 a -0.1111111111111111)
   (/ rand (sqrt (fabs (- 0.3333333333333333 a)))))
  (if (<= a 10000.0)
    (/ rand (sqrt (/ 9.0 a)))
    (fma
     (/ (- a 0.3333333333333333) (sqrt (fma 9.0 a -3.0)))
     rand
     (- a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= -2e-310) {
		tmp = fma(0.3333333333333333, a, -0.1111111111111111) * (rand / sqrt(fabs((0.3333333333333333 - a))));
	} else if (a <= 10000.0) {
		tmp = rand / sqrt((9.0 / a));
	} else {
		tmp = fma(((a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= -2e-310)
		tmp = Float64(fma(0.3333333333333333, a, -0.1111111111111111) * Float64(rand / sqrt(abs(Float64(0.3333333333333333 - a)))));
	elseif (a <= 10000.0)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	else
		tmp = fma(Float64(Float64(a - 0.3333333333333333) / sqrt(fma(9.0, a, -3.0))), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, -2e-310], N[(N[(0.3333333333333333 * a + -0.1111111111111111), $MachinePrecision] * N[(rand / N[Sqrt[N[Abs[N[(0.3333333333333333 - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 10000.0], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.3333333333333333), $MachinePrecision] / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9999999999999939e-310

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. sqrt-prod N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. fabs-sub N/A

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

      18. frac-times N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 14.3%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \color{blue}{\frac{rand}{\sqrt{\left|0.3333333333333333 - a\right|}}} \]

   if -1.9999999999999939e-310 < a < 1e4

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.2%

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

   9. Applied rewrites 13.2%

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

   if 1e4 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

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

Alternative 7: 32.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left|0.3333333333333333 - a\right|}\\ \mathbf{if}\;a \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \frac{rand}{t\_0}\\ \mathbf{elif}\;a \leq 10000:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{t\_0}, rand, -1\right) \cdot \left(0.3333333333333333 - a\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (let* ((t_0 (sqrt (fabs (- 0.3333333333333333 a)))))
  (if (<= a -2e-310)
    (* (fma 0.3333333333333333 a -0.1111111111111111) (/ rand t_0))
    (if (<= a 10000.0)
      (/ rand (sqrt (/ 9.0 a)))
      (*
       (fma (/ -0.3333333333333333 t_0) rand -1.0)
       (- 0.3333333333333333 a))))))

C

double code(double a, double rand) {
	double t_0 = sqrt(fabs((0.3333333333333333 - a)));
	double tmp;
	if (a <= -2e-310) {
		tmp = fma(0.3333333333333333, a, -0.1111111111111111) * (rand / t_0);
	} else if (a <= 10000.0) {
		tmp = rand / sqrt((9.0 / a));
	} else {
		tmp = fma((-0.3333333333333333 / t_0), rand, -1.0) * (0.3333333333333333 - a);
	}
	return tmp;
}

Julia

function code(a, rand)
	t_0 = sqrt(abs(Float64(0.3333333333333333 - a)))
	tmp = 0.0
	if (a <= -2e-310)
		tmp = Float64(fma(0.3333333333333333, a, -0.1111111111111111) * Float64(rand / t_0));
	elseif (a <= 10000.0)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	else
		tmp = Float64(fma(Float64(-0.3333333333333333 / t_0), rand, -1.0) * Float64(0.3333333333333333 - a));
	end
	return tmp
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(0.3333333333333333 - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -2e-310], N[(N[(0.3333333333333333 * a + -0.1111111111111111), $MachinePrecision] * N[(rand / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 10000.0], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.3333333333333333 / t$95$0), $MachinePrecision] * rand + -1.0), $MachinePrecision] * N[(0.3333333333333333 - a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9999999999999939e-310

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. sqrt-prod N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. fabs-sub N/A

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

      18. frac-times N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 14.3%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \color{blue}{\frac{rand}{\sqrt{\left|0.3333333333333333 - a\right|}}} \]

   if -1.9999999999999939e-310 < a < 1e4

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.2%

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

   9. Applied rewrites 13.2%

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

   if 1e4 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. distribute-rgt-neg-out N/A

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

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

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. sub-negate-rev N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 26.9%

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

Alternative 8: 32.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \frac{rand}{\sqrt{\left|0.3333333333333333 - a\right|}}\\ \mathbf{elif}\;a \leq 10000:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}} - -1\right) \cdot \left(a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a -2e-310)
  (*
   (fma 0.3333333333333333 a -0.1111111111111111)
   (/ rand (sqrt (fabs (- 0.3333333333333333 a)))))
  (if (<= a 10000.0)
    (/ rand (sqrt (/ 9.0 a)))
    (*
     (- (/ rand (sqrt (fma 9.0 a -3.0))) -1.0)
     (- a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= -2e-310) {
		tmp = fma(0.3333333333333333, a, -0.1111111111111111) * (rand / sqrt(fabs((0.3333333333333333 - a))));
	} else if (a <= 10000.0) {
		tmp = rand / sqrt((9.0 / a));
	} else {
		tmp = ((rand / sqrt(fma(9.0, a, -3.0))) - -1.0) * (a - 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= -2e-310)
		tmp = Float64(fma(0.3333333333333333, a, -0.1111111111111111) * Float64(rand / sqrt(abs(Float64(0.3333333333333333 - a)))));
	elseif (a <= 10000.0)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	else
		tmp = Float64(Float64(Float64(rand / sqrt(fma(9.0, a, -3.0))) - -1.0) * Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, -2e-310], N[(N[(0.3333333333333333 * a + -0.1111111111111111), $MachinePrecision] * N[(rand / N[Sqrt[N[Abs[N[(0.3333333333333333 - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 10000.0], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(rand / N[Sqrt[N[(9.0 * a + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9999999999999939e-310

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. sqrt-prod N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. fabs-sub N/A

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

      18. frac-times N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 14.3%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \color{blue}{\frac{rand}{\sqrt{\left|0.3333333333333333 - a\right|}}} \]

   if -1.9999999999999939e-310 < a < 1e4

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.2%

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

   9. Applied rewrites 13.2%

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

   if 1e4 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

Alternative 9: 31.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \frac{rand}{\sqrt{\left|0.3333333333333333 - a\right|}}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\left|a\right|}{9}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a -2e-310)
  (*
   (fma 0.3333333333333333 a -0.1111111111111111)
   (/ rand (sqrt (fabs (- 0.3333333333333333 a)))))
  (if (<= a 3.2e+26)
    (/ rand (sqrt (/ 9.0 a)))
    (fma (sqrt (/ (fabs a) 9.0)) rand (- a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= -2e-310) {
		tmp = fma(0.3333333333333333, a, -0.1111111111111111) * (rand / sqrt(fabs((0.3333333333333333 - a))));
	} else if (a <= 3.2e+26) {
		tmp = rand / sqrt((9.0 / a));
	} else {
		tmp = fma(sqrt((fabs(a) / 9.0)), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= -2e-310)
		tmp = Float64(fma(0.3333333333333333, a, -0.1111111111111111) * Float64(rand / sqrt(abs(Float64(0.3333333333333333 - a)))));
	elseif (a <= 3.2e+26)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	else
		tmp = fma(sqrt(Float64(abs(a) / 9.0)), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, -2e-310], N[(N[(0.3333333333333333 * a + -0.1111111111111111), $MachinePrecision] * N[(rand / N[Sqrt[N[Abs[N[(0.3333333333333333 - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.2e+26], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[Abs[a], $MachinePrecision] / 9.0), $MachinePrecision]], $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9999999999999939e-310

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. sqrt-prod N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. fabs-sub N/A

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

      18. frac-times N/A

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

      19. lower-*.f64 N/A

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

   8. Applied rewrites 14.3%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, a, -0.1111111111111111\right) \cdot \color{blue}{\frac{rand}{\sqrt{\left|0.3333333333333333 - a\right|}}} \]

   if -1.9999999999999939e-310 < a < 3.2000000000000003e26

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.2%

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

   9. Applied rewrites 13.2%

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

   if 3.2000000000000003e26 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 25.5%

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

   6. Applied rewrites 25.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. div-flip-rev N/A

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

      8. metadata-eval N/A

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

      9. sqrt-undiv N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-fabs.f64 26.7%

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

   8. Applied rewrites 26.7%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{\left|a\right|}{9}}, rand, a - 0.3333333333333333\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 29.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 2.05 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.1111111111111111}{a - -0.3333333333333333} \cdot 1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\left|a\right|}{9}}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a 2.05e-305)
  (* (/ -0.1111111111111111 (- a -0.3333333333333333)) 1.0)
  (if (<= a 3.2e+26)
    (/ rand (sqrt (/ 9.0 a)))
    (fma (sqrt (/ (fabs a) 9.0)) rand (- a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= 2.05e-305) {
		tmp = (-0.1111111111111111 / (a - -0.3333333333333333)) * 1.0;
	} else if (a <= 3.2e+26) {
		tmp = rand / sqrt((9.0 / a));
	} else {
		tmp = fma(sqrt((fabs(a) / 9.0)), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= 2.05e-305)
		tmp = Float64(Float64(-0.1111111111111111 / Float64(a - -0.3333333333333333)) * 1.0);
	elseif (a <= 3.2e+26)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	else
		tmp = fma(sqrt(Float64(abs(a) / 9.0)), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, 2.05e-305], N[(N[(-0.1111111111111111 / N[(a - -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[a, 3.2e+26], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[Abs[a], $MachinePrecision] / 9.0), $MachinePrecision]], $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.0500000000000001e-305

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. remove-sound-/ N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. add-flip N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{9}}}{a - -0.3333333333333333} \cdot 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 3.3%

      \[\leadsto \frac{\color{blue}{-0.1111111111111111}}{a - -0.3333333333333333} \cdot 1 \]

   if 2.0500000000000001e-305 < a < 3.2000000000000003e26

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.0%

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in rand around inf

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

      1. metadata-eval N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 7.6%

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

   6. Applied rewrites 7.6%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 13.2%

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

   9. Applied rewrites 13.2%

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

   if 3.2000000000000003e26 < a

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-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. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 25.0%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 25.5%

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

   6. Applied rewrites 25.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. div-flip-rev N/A

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

      8. metadata-eval N/A

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

      9. sqrt-undiv N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-fabs.f64 26.7%

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

   8. Applied rewrites 26.7%

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{\left|a\right|}{9}}, rand, a - 0.3333333333333333\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 29.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 2.05 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.1111111111111111}{a - -0.3333333333333333} \cdot 1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot \sqrt{\left|a\right|}, rand, a - 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (if (<= a 2.05e-305)
  (* (/ -0.1111111111111111 (- a -0.3333333333333333)) 1.0)
  (if (<= a 3.2e+26)
    (/ rand (sqrt (/ 9.0 a)))
    (fma
     (* 0.3333333333333333 (sqrt (fabs a)))
     rand
     (- a 0.3333333333333333)))))

C

double code(double a, double rand) {
	double tmp;
	if (a <= 2.05e-305) {
		tmp = (-0.1111111111111111 / (a - -0.3333333333333333)) * 1.0;
	} else if (a <= 3.2e+26) {
		tmp = rand / sqrt((9.0 / a));
	} else {
		tmp = fma((0.3333333333333333 * sqrt(fabs(a))), rand, (a - 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(a, rand)
	tmp = 0.0
	if (a <= 2.05e-305)
		tmp = Float64(Float64(-0.1111111111111111 / Float64(a - -0.3333333333333333)) * 1.0);
	elseif (a <= 3.2e+26)
		tmp = Float64(rand / sqrt(Float64(9.0 / a)));
	else
		tmp = fma(Float64(0.3333333333333333 * sqrt(abs(a))), rand, Float64(a - 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[a_, rand_] := If[LessEqual[a, 2.05e-305], N[(N[(-0.1111111111111111 / N[(a - -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[a, 3.2e+26], N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand + N[(a - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.0500000000000001e-305

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 18.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(a - \color{blue}{\frac{1}{3}}\right) \cdot 1 \]

      4. flip-- N/A

         \[\leadsto \color{blue}{\frac{a \cdot a - \frac{1}{3} \cdot \frac{1}{3}}{a + \frac{1}{3}}} \cdot 1 \]

      5. +-commutative N/A

         \[\leadsto \frac{a \cdot a - \frac{1}{3} \cdot \frac{1}{3}}{\color{blue}{\frac{1}{3} + a}} \cdot 1 \]

      6. remove-sound-/ N/A

         \[\leadsto \color{blue}{\frac{a \cdot a - \frac{1}{3} \cdot \frac{1}{3}}{\frac{1}{3} + a}} \cdot 1 \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot a - \frac{1}{3} \cdot \frac{1}{3}}{\frac{1}{3} + a}} \cdot 1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{a \cdot a + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{3}}}{\frac{1}{3} + a} \cdot 1 \]

      9. metadata-eval N/A

         \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{3}} \cdot \frac{1}{3}}{\frac{1}{3} + a} \cdot 1 \]

      10. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{-1}{9}}}{\frac{1}{3} + a} \cdot 1 \]

      11. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \color{blue}{\frac{\frac{-1}{3}}{3}}}{\frac{1}{3} + a} \cdot 1 \]

      12. metadata-eval N/A

          \[\leadsto \frac{a \cdot a + \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{3}}{\frac{1}{3} + a} \cdot 1 \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, a, \frac{\mathsf{neg}\left(\frac{1}{3}\right)}{3}\right)}}{\frac{1}{3} + a} \cdot 1 \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{\color{blue}{\frac{-1}{3}}}{3}\right)}{\frac{1}{3} + a} \cdot 1 \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \color{blue}{\frac{-1}{9}}\right)}{\frac{1}{3} + a} \cdot 1 \]

      16. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\color{blue}{a + \frac{1}{3}}} \cdot 1 \]

      17. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\color{blue}{a - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}} \cdot 1 \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, a, \frac{-1}{9}\right)}{\color{blue}{a - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}} \cdot 1 \]

      19. metadata-eval 10.8%

          \[\leadsto \frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - \color{blue}{-0.3333333333333333}} \cdot 1 \]

   6. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a, -0.1111111111111111\right)}{a - -0.3333333333333333}} \cdot 1 \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{9}}}{a - -0.3333333333333333} \cdot 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 3.3%

      \[\leadsto \frac{\color{blue}{-0.1111111111111111}}{a - -0.3333333333333333} \cdot 1 \]

   if 2.0500000000000001e-305 < a < 3.2000000000000003e26

   1. Initial program 25.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)} \]

      3. lower-*.f64 25.0%

         \[\leadsto \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)} \]

   3. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(\frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}} - -1\right) \cdot \left(a - 0.3333333333333333\right)} \]

   4. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\color{blue}{\sqrt{9 \cdot a - 3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{\color{blue}{9 \cdot a - 3}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - \color{blue}{3}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      8. lower-*.f64 7.6%

         \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot a - 3}} \]

   6. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot a - 3}}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      3. lower-/.f64 13.2%

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

   9. Applied rewrites 13.2%

      \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]

   if 3.2000000000000003e26 < a

   1. Initial program 25.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand + 1\right)} \]

      4. distribute-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. *-commutative N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} + 1 \cdot \left(a - \frac{1}{3}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} + 1 \cdot \left(a - \frac{1}{3}\right) \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand} + 1 \cdot \left(a - \frac{1}{3}\right) \]

      8. *-lft-identity N/A

         \[\leadsto \left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand + \color{blue}{\left(a - \frac{1}{3}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, a - \frac{1}{3}\right)} \]

   3. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a - 0.3333333333333333}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}}, rand, a - 0.3333333333333333\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{9}{a}}}}, rand, a - 0.3333333333333333\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{9}{a}}}}, rand, a - \frac{1}{3}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\frac{9}{a}}}, rand, a - \frac{1}{3}\right) \]

      3. lower-/.f64 25.5%

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\frac{9}{a}}}, rand, a - 0.3333333333333333\right) \]

   6. Applied rewrites 25.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\sqrt{\frac{9}{a}}}}, rand, a - 0.3333333333333333\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{\frac{9}{a}}}}, rand, a - \frac{1}{3}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\frac{9}{a}}}, rand, a - \frac{1}{3}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{\frac{9}{a}}}, rand, a - \frac{1}{3}\right) \]

      4. sqrt-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\sqrt{\left|9\right|}}{\color{blue}{\sqrt{\left|a\right|}}}}, rand, a - \frac{1}{3}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{\sqrt{9}}{\sqrt{\left|\color{blue}{a}\right|}}}, rand, a - \frac{1}{3}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{3}{\sqrt{\color{blue}{\left|a\right|}}}}, rand, a - \frac{1}{3}\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot \color{blue}{\sqrt{\left|a\right|}}, rand, a - \frac{1}{3}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot \sqrt{\color{blue}{\left|a\right|}}, rand, a - \frac{1}{3}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot \color{blue}{\sqrt{\left|a\right|}}, rand, a - \frac{1}{3}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot \sqrt{\left|a\right|}, rand, a - \frac{1}{3}\right) \]

      11. lower-fabs.f64 26.7%

          \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot \sqrt{\left|a\right|}, rand, a - 0.3333333333333333\right) \]

   8. Applied rewrites 26.7%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot \color{blue}{\sqrt{\left|a\right|}}, rand, a - 0.3333333333333333\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 24.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{rand}{\sqrt{\frac{9}{a}}}\\ \mathbf{if}\;rand \leq -8.9 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 4.2 \cdot 10^{+56}:\\ \;\;\;\;\left(a \cdot 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a rand)
  :precision binary64
  (let* ((t_0 (/ rand (sqrt (/ 9.0 a)))))
  (if (<= rand -8.9e+46)
    t_0
    (if (<= rand 4.2e+56) (* (* a 1.0) 1.0) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand / sqrt((9.0 / a));
	double tmp;
	if (rand <= -8.9e+46) {
		tmp = t_0;
	} else if (rand <= 4.2e+56) {
		tmp = (a * 1.0) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = rand / sqrt((9.0d0 / a))
    if (rand <= (-8.9d+46)) then
        tmp = t_0
    else if (rand <= 4.2d+56) then
        tmp = (a * 1.0d0) * 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand / Math.sqrt((9.0 / a));
	double tmp;
	if (rand <= -8.9e+46) {
		tmp = t_0;
	} else if (rand <= 4.2e+56) {
		tmp = (a * 1.0) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand / math.sqrt((9.0 / a))
	tmp = 0
	if rand <= -8.9e+46:
		tmp = t_0
	elif rand <= 4.2e+56:
		tmp = (a * 1.0) * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand / sqrt(Float64(9.0 / a)))
	tmp = 0.0
	if (rand <= -8.9e+46)
		tmp = t_0;
	elseif (rand <= 4.2e+56)
		tmp = Float64(Float64(a * 1.0) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand / sqrt((9.0 / a));
	tmp = 0.0;
	if (rand <= -8.9e+46)
		tmp = t_0;
	elseif (rand <= 4.2e+56)
		tmp = (a * 1.0) * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand / N[Sqrt[N[(9.0 / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -8.9e+46], t$95$0, If[LessEqual[rand, 4.2e+56], N[(N[(a * 1.0), $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -8.8999999999999997e46 or 4.2000000000000003e56 < rand

   1. Initial program 25.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)} \]

      3. lower-*.f64 25.0%

         \[\leadsto \color{blue}{\left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)} \]

   3. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(\frac{rand}{\sqrt{\mathsf{fma}\left(9, a, -3\right)}} - -1\right) \cdot \left(a - 0.3333333333333333\right)} \]

   4. Taylor expanded in rand around inf

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\color{blue}{\sqrt{9 \cdot a - 3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{\color{blue}{9 \cdot a - 3}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - \color{blue}{3}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{rand \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9 \cdot a - 3}} \]

      8. lower-*.f64 7.6%

         \[\leadsto \frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot a - 3}} \]

   6. Applied rewrites 7.6%

      \[\leadsto \color{blue}{\frac{rand \cdot \left(a - 0.3333333333333333\right)}{\sqrt{9 \cdot a - 3}}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

      3. lower-/.f64 13.2%

         \[\leadsto \frac{rand}{\sqrt{\frac{9}{a}}} \]

   9. Applied rewrites 13.2%

      \[\leadsto \frac{rand}{\color{blue}{\sqrt{\frac{9}{a}}}} \]

   if -8.8999999999999997e46 < rand < 4.2000000000000003e56

   1. Initial program 25.0%

      \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

   2. Taylor expanded in a around inf

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 18.0%

      \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\left(a \cdot \left(1 - \frac{1}{3} \cdot \frac{1}{a}\right)\right)} \cdot 1 \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(a \cdot \left(1 - \frac{1}{3} \cdot \frac{\color{blue}{1}}{a}\right)\right) \cdot 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \color{blue}{\left(1 - \frac{1}{3} \cdot \frac{1}{a}\right)}\right) \cdot 1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(a \cdot \left(1 - \color{blue}{\frac{1}{3} \cdot \frac{1}{a}}\right)\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(1 - \frac{1}{3} \cdot \color{blue}{\frac{1}{a}}\right)\right) \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(a \cdot \left(1 - \frac{1}{3} \cdot \frac{\color{blue}{1}}{a}\right)\right) \cdot 1 \]

      6. lower-/.f64 18.0%

         \[\leadsto \left(a \cdot \left(1 - 0.3333333333333333 \cdot \frac{1}{\color{blue}{a}}\right)\right) \cdot 1 \]

   7. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\left(a \cdot \left(1 - 0.3333333333333333 \cdot \frac{1}{a}\right)\right)} \cdot 1 \]

   8. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot 1\right) \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 18.9%

       \[\leadsto \left(a \cdot 1\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 18.9% accurate, 4.5× speedup

Math

\[\left(a \cdot 1\right) \cdot 1 \]

FPCore

(FPCore (a rand)
  :precision binary64
  (* (* a 1.0) 1.0))

C

double code(double a, double rand) {
	return (a * 1.0) * 1.0;
}

Fortran

real(8) function code(a, rand)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a * 1.0d0) * 1.0d0
end function

Java

public static double code(double a, double rand) {
	return (a * 1.0) * 1.0;
}

Python

def code(a, rand):
	return (a * 1.0) * 1.0

Julia

function code(a, rand)
	return Float64(Float64(a * 1.0) * 1.0)
end

MATLAB

function tmp = code(a, rand)
	tmp = (a * 1.0) * 1.0;
end

Wolfram

code[a_, rand_] := N[(N[(a * 1.0), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 25.0%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

2. Taylor expanded in a around inf

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 18.0%

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]

5. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\left(a \cdot \left(1 - \frac{1}{3} \cdot \frac{1}{a}\right)\right)} \cdot 1 \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(a \cdot \left(1 - \frac{1}{3} \cdot \frac{\color{blue}{1}}{a}\right)\right) \cdot 1 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(a \cdot \color{blue}{\left(1 - \frac{1}{3} \cdot \frac{1}{a}\right)}\right) \cdot 1 \]

   3. lower--.f64 N/A

      \[\leadsto \left(a \cdot \left(1 - \color{blue}{\frac{1}{3} \cdot \frac{1}{a}}\right)\right) \cdot 1 \]

   4. lower-*.f64 N/A

      \[\leadsto \left(a \cdot \left(1 - \frac{1}{3} \cdot \color{blue}{\frac{1}{a}}\right)\right) \cdot 1 \]

   5. metadata-eval N/A

      \[\leadsto \left(a \cdot \left(1 - \frac{1}{3} \cdot \frac{\color{blue}{1}}{a}\right)\right) \cdot 1 \]

   6. lower-/.f64 18.0%

      \[\leadsto \left(a \cdot \left(1 - 0.3333333333333333 \cdot \frac{1}{\color{blue}{a}}\right)\right) \cdot 1 \]

7. Applied rewrites 18.0%

   \[\leadsto \color{blue}{\left(a \cdot \left(1 - 0.3333333333333333 \cdot \frac{1}{a}\right)\right)} \cdot 1 \]

8. Taylor expanded in a around inf

   \[\leadsto \left(a \cdot 1\right) \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 18.9%

    \[\leadsto \left(a \cdot 1\right) \cdot 1 \]
11. Add Preprocessing

Alternative 14: 18.0% accurate, 4.7× speedup

Math

\[\left(a - 0.3333333333333333\right) \cdot 1 \]

FPCore

(FPCore (a rand)
  :precision binary64
  (* (- a 0.3333333333333333) 1.0))

C

double code(double a, double rand) {
	return (a - 0.3333333333333333) * 1.0;
}

Fortran

real(8) function code(a, rand)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a - 0.3333333333333333d0) * 1.0d0
end function

Java

public static double code(double a, double rand) {
	return (a - 0.3333333333333333) * 1.0;
}

Python

def code(a, rand):
	return (a - 0.3333333333333333) * 1.0

Julia

function code(a, rand)
	return Float64(Float64(a - 0.3333333333333333) * 1.0)
end

MATLAB

function tmp = code(a, rand)
	tmp = (a - 0.3333333333333333) * 1.0;
end

Wolfram

code[a_, rand_] := N[(N[(a - 0.3333333333333333), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 25.0%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

2. Taylor expanded in a around inf

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 18.0%

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(a - \color{blue}{\frac{1}{3}}\right) \cdot 1 \]

   2. metadata-eval 18.0%

      \[\leadsto \left(a - \color{blue}{0.3333333333333333}\right) \cdot 1 \]

6. Applied rewrites 18.0%

   \[\leadsto \left(a - \color{blue}{0.3333333333333333}\right) \cdot 1 \]
7. Add Preprocessing

Alternative 15: 2.7% accurate, 7.8× speedup

Math

\[-0.3333333333333333 \cdot 1 \]

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)
use fmin_fmax_functions
    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 25.0%

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

2. Taylor expanded in a around inf

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 18.0%

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \color{blue}{1} \]

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{3}} \cdot 1 \]
6. Step-by-step derivation

7. Applied rewrites 2.7%

   \[\leadsto \color{blue}{-0.3333333333333333} \cdot 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(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))))