Octave 3.8, oct_fill_randg

Percentage Accurate: 99.7% → 99.8%

Time: 9.8s

Alternatives: 9

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

   1. metadata-eval N/A

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

   2. lower--.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 3: 91.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{if}\;rand \leq -6.7 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 9 \cdot 10^{+58}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* (* rand 0.3333333333333333) (sqrt a))))
   (if (<= rand -6.7e+62)
     t_0
     (if (<= rand 9e+58) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = (rand * 0.3333333333333333) * sqrt(a);
	double tmp;
	if (rand <= -6.7e+62) {
		tmp = t_0;
	} else if (rand <= 9e+58) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (rand * 0.3333333333333333d0) * sqrt(a)
    if (rand <= (-6.7d+62)) then
        tmp = t_0
    else if (rand <= 9d+58) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = (rand * 0.3333333333333333) * Math.sqrt(a);
	double tmp;
	if (rand <= -6.7e+62) {
		tmp = t_0;
	} else if (rand <= 9e+58) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = (rand * 0.3333333333333333) * math.sqrt(a)
	tmp = 0
	if rand <= -6.7e+62:
		tmp = t_0
	elif rand <= 9e+58:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(Float64(rand * 0.3333333333333333) * sqrt(a))
	tmp = 0.0
	if (rand <= -6.7e+62)
		tmp = t_0;
	elseif (rand <= 9e+58)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = (rand * 0.3333333333333333) * sqrt(a);
	tmp = 0.0;
	if (rand <= -6.7e+62)
		tmp = t_0;
	elseif (rand <= 9e+58)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(N[(rand * 0.3333333333333333), $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -6.7e+62], t$95$0, If[LessEqual[rand, 9e+58], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -6.6999999999999997e62 or 8.9999999999999996e58 < rand

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 97.8

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

   7. Simplified 97.8%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 85.5

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

   10. Simplified 85.5%

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

   if -6.6999999999999997e62 < rand < 8.9999999999999996e58

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 95.9

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

   7. Simplified 95.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;rand \leq -6.7 \cdot 10^{+62}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \mathbf{elif}\;rand \leq 9 \cdot 10^{+58}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(rand \cdot 0.3333333333333333\right) \cdot \sqrt{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := rand \cdot \left(0.3333333333333333 \cdot \sqrt{a}\right)\\ \mathbf{if}\;rand \leq -6.7 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;rand \leq 9 \cdot 10^{+58}:\\ \;\;\;\;a + -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a rand)
 :precision binary64
 (let* ((t_0 (* rand (* 0.3333333333333333 (sqrt a)))))
   (if (<= rand -6.7e+62)
     t_0
     (if (<= rand 9e+58) (+ a -0.3333333333333333) t_0))))

C

double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * sqrt(a));
	double tmp;
	if (rand <= -6.7e+62) {
		tmp = t_0;
	} else if (rand <= 9e+58) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    real(8) :: t_0
    real(8) :: tmp
    t_0 = rand * (0.3333333333333333d0 * sqrt(a))
    if (rand <= (-6.7d+62)) then
        tmp = t_0
    else if (rand <= 9d+58) then
        tmp = a + (-0.3333333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double rand) {
	double t_0 = rand * (0.3333333333333333 * Math.sqrt(a));
	double tmp;
	if (rand <= -6.7e+62) {
		tmp = t_0;
	} else if (rand <= 9e+58) {
		tmp = a + -0.3333333333333333;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, rand):
	t_0 = rand * (0.3333333333333333 * math.sqrt(a))
	tmp = 0
	if rand <= -6.7e+62:
		tmp = t_0
	elif rand <= 9e+58:
		tmp = a + -0.3333333333333333
	else:
		tmp = t_0
	return tmp

Julia

function code(a, rand)
	t_0 = Float64(rand * Float64(0.3333333333333333 * sqrt(a)))
	tmp = 0.0
	if (rand <= -6.7e+62)
		tmp = t_0;
	elseif (rand <= 9e+58)
		tmp = Float64(a + -0.3333333333333333);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, rand)
	t_0 = rand * (0.3333333333333333 * sqrt(a));
	tmp = 0.0;
	if (rand <= -6.7e+62)
		tmp = t_0;
	elseif (rand <= 9e+58)
		tmp = a + -0.3333333333333333;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, rand_] := Block[{t$95$0 = N[(rand * N[(0.3333333333333333 * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[rand, -6.7e+62], t$95$0, If[LessEqual[rand, 9e+58], N[(a + -0.3333333333333333), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if rand < -6.6999999999999997e62 or 8.9999999999999996e58 < rand

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 97.8

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

   7. Simplified 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in a around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sqrt.f64 85.4

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

   12. Simplified 85.4%

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

   if -6.6999999999999997e62 < rand < 8.9999999999999996e58

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-to-exp N/A

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

      6. pow-to-exp N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in rand around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 95.9

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

   7. Simplified 95.9%

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

Alternative 5: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*l/ N/A

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

   14. associate-*r/ N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in rand around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. lower-+.f64 99.8

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

7. Simplified 99.8%

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

Alternative 6: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 98.9

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

9. Simplified 98.9%

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

Alternative 7: 97.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 98.2

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

7. Simplified 98.2%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

       \[\leadsto \mathsf{fma}\left(rand \cdot \frac{1}{3}, \color{blue}{{\left(\frac{1}{a}\right)}^{\frac{1}{2}}} \cdot a, a\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(rand \cdot \frac{1}{3}, {\color{blue}{\left(\frac{1}{a}\right)}}^{\frac{1}{2}} \cdot a, a\right) \]

   13. inv-pow N/A

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

   14. pow-pow N/A

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

   15. pow-plus N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(rand \cdot \frac{1}{3}, {a}^{\color{blue}{\frac{1}{2}}}, a\right) \]

   18. pow1/2 N/A

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

   19. lower-sqrt.f64 98.3

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

9. Applied egg-rr 98.3%

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

Alternative 8: 62.7% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ a + -0.3333333333333333 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 60.9

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

7. Simplified 60.9%

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

Alternative 9: 1.5% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, rand):
	return -0.3333333333333333

Julia

function code(a, rand)
	return -0.3333333333333333
end

MATLAB

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

Wolfram

code[a_, rand_] := -0.3333333333333333

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-to-exp N/A

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

   6. pow-to-exp N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-/.f64 N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in rand around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 60.9

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

7. Simplified 60.9%

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

8. Taylor expanded in a around 0

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

10. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024219 
(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))))